It's OK to tell them what is and what is not a dungeon

My guess is that the problem comes from the assumption that you don't want to tell them what is and what is not a dungeon in case it breaks the immersion and the seamless transition between walking into a dungeon from the "overworld" and exploring the dungeon.

For example; in, say, a Bethesda game like Oblivion or Skyrim, there's literally a loading screen between dungeons and overworld sections. In D&D there is not such a loading screen and you can therefore make everywhere seem like one big world rather than an "area" that must be "loaded" into memory or whatever.

However, in my experience, players can still tell what is and what is not a dungeon, at least once they're already in the dungeon, they'll know they're now in a dungeon.

Tell them when to draw a map and when not to draw

Funnily enough, I have recently had new players playing LMoP and the concept of map drawing had to be introduced to them. At first, they had a hard time visualising what I was telling them, but I ended up just drawing the map for them myself, although this was largely because I did such a poor job of explaining area 2 of that first dungeon (from memory, I think it was area 2; just outside the cave where there are two goblins hiding between bushes, and a stream coming from the mouth of the cave that is the dungeon? I've run this adventure twice and both times I completely failed to describe that scene accurately; it's just that one area, I'm fine with the rest of the dungeon and all the other dungeons in the adventure!)

Anyway, the next time we encountered a dungeon (I think it was the Redbrand Hideout), they expected me to draw the map again, but I recommended they do it, partly so they could confirm that they were understanding my descriptions correctly, but I also told them I was just being nice before and that this is actually something they should be doing, not me.

Since they can tell what is and what is not a dungeon, they have now drawn maps when we enter a dungeon and not during any other point.

Summary

In other words, I don't think they'll hear

"You should draw a map right now"

if you give them context (i.e. you should draw maps when you're in a dungeon) and I think they'll understand when that applies and when it doesn't.

Don't have the players draw maps (at least not for the reasons you list)

From everything I have experienced with players new and old, having players make the maps has never worked well and I would recommend not doing it at your table.

Firstly though, I think there are some serious issues with how you are approaching this question from even a conceptual level.

Your players may not want to draw maps; you need to ask them first

You say:

For me it bring some realistic side (maybe my players
will not like that, but that's not the point here) and I like it
pretty much.

Right off the bat, this is a bad attitude for you to have as a DM and will lead to problems at the table. You, the DM, are not the only one at the table who matters. Specifically, you need to take into consideration the wants and needs of the whole table, not just for you. So whether or not the players will like it is very much relevant. You need to talk to the players before going any further and figure out if this is even something they remotely want.

I'm a player who would vehemently say no to this if asked for my preference. Some people I play with feel otherwise. You shouldn't forcibly introduce a mechanic at your table that will reduce the fun at your table. Talk to your players.

You are crossing the wires of in-game and out-of-game issues

I'm talking about players physically drawing a map, not only their
characters.

You say that, but then most of your question is dedicated to introducing how this real-life map will affect the in-game world and the PCs. This is a mistake you need to sort out. Unless you have player buy-in, real-world challenges should not affect the abilities or successes of the PCs in-game. Players play games because they want to do stuff that they are not capable or wanting to do in real life. Making game challenges depend on real-world successes means that players who aren't good at that task get their in-game characters punished unjustly.

If this is your main motivation for doing this, please read the rest of this section and then seriously reconsider this idea.

I think it's much realistic and interesting to test the real memory of the players instead of assuming their characters remember everything they see.

You don't have to assume that a character remembers things perfectly, but the game already has mechanics in it to address that (read: skill checks).

What interesting results could possibly result from a player misremembering something that their character did not? The only thing I could see resulting from that is frustration and upsetedness at the table.

The characters can draw maps just fine without having to have the players do so in real life. You can even simulate this with a survival check (or a series of them).

Think of it this way, what if one of the PCs was a map-drawing master. An explorer by trade with an unerring sense of direction (a Ranger, say). What sense does it make to make that player's map-making success depend on the result of the players' skills?

It's also a way to give advantages to players who pay attention to description and may make it easier to "punish" distracted players (don't worry for them, I'm not that evil).

If you have issues with players not paying attention, you should address this issue directly and not punish the rest of the players/the in-game characters for it. Solve out-of-game issues with out-of-game solutions.

There's no problem to solve here

As I wrote above, I would enjoy having them invest into drawing a map. But it's also for when they we'll need a map to don't get lost in a very big maze or else, for example if we play Waterdeep Dungeon of the Mad Mage.

As I mentioned above, you should not be crossing in-game and out-of-game challenges. But more than that, you don't seem to have a compelling reason to need this beyond the fact that you (and currently only you) want it.

I'm going to list some of the issues you will face should you try to implement this and the complications that will result. In the end, I believe these downsides far outweigh any pleasure you think you'll get from having the players do this task (even apart from whether they want to or not).

Downsides to having players draw maps

Every one of these issues is one that has repeatedly come up at my tables throughout the years.

It takes a lot of time

I'm not joking when I say this, even in a simple dungeon, having the players draw maps can and likely will slow progress to a crawl. This is because it requires the DM to give all the information about the room (dimensions, locations, doorways, location on the map, etc.) while the player is sketching. During this time the DM could instead be describing the room narratively, interacting with the PCs with their actions in the room, looking at encounter information, reading ahead, or literally any other productive task. Instead, they are describing a map which they have drawn out in front of them to someone else who can't see that map. Even if the players hear and translate that information perfectly onto the map, this still takes precious time away from a session that could be used for other things.

At tables I've played at, the time it takes to draw the map out in real time has been a large source of frustration even among people that enjoy making the map.

Mistakes will be made

No matter how obvious you think your description of the room is, the map-drawers will accidentally get it wrong. A lot. They'll do things like accidentally flip width vs height, put the door on the wrong side, attach the room on the wrong wall, etc. And when they do you have to spend even more time to have them erase it and then correct it. This happened once in an error so bad the DM had to take it home and sketch a decent portion of the map again.

And that is if you manage to notice the error. Many times I've had DMs not notice until hours later that a mistake was made on the map and now suddenly this room won't fit! You think this won't happen to you, but it will. DMs are busy and even if you take (even more) time to check over a drawing, you too will miss errors (a common source of DM error: the fact that you are looking at their map "upside down").

Sometimes, you will end up just giving up and (re)drawing things out for the players because of misunderstandings.

It's boring for most, if not all, of the players

You mentioned worrying about players not paying attention before. Well this is a great way to get them to pay even less attention. It only takes one person to draw a map and it isn't particularly a thrilling process. Expect phones to come out and people to leave and get snacks or drinks while this occurs. This is the opposite of one of the things you mentioned wanting to do which is to keep their attention. At most, you get one player's attention and lose the rest of them while things are worked out. I have personally experienced it both as the checking out player and observing others doing the same.

New players have enough to keep track of

You mention this is for new players. This is going to be a challenge to them but they likely will already have their hands full trying to learn and manage their characters and new abilities and roleplaying. Having them take time away from that can lead to less-prepared and engaged players.

Conclusion

I don't recommend doing this because it is a whole lot of hassle for almost (if not entirely) no positive gain in fun for the table.

Here's what you need to do to even have a chance at this being successful:

• Reevaluate why you want this to happen. Challenging the players instead of the characters is not a good reason and will likely not resonate with your players and lead to frustration. If you are going to introduce this, you need a good reason, one that is fun for the whole table.


• Talk to your players. If they aren't interested, this idea is a non-starter regardless of what you personally want. If they are interested, you need to talk to them explicitly and in-detail about how this will work and make sure they agree to it and that it seems fair.


• Once you've done that you are going to need to tackle the practical
  issues I mentioned above. Problem is, I've not seen any great solutions. If you don't tackle them, you need to prepare for longer sessions and less engaged players.

I usually just draw the map part by part for them, as far as they discover it. Depending on the paper you have (mainly, is it big and does it have squares), this might even be useful for combat. This means your players won't need to draw anything. Sure, it costs me time to draw it out, but it also helps them understand what kind of area they are in a bit more. (I am also currently running Lost Mines, 2 new players, 1 with some experience and 1 experienced player. Even though I can't really draw that well, they still enjoy that I draw things out for them)

EDIT: I think this might be more useful, especially since this adventure won't see the players navigate dungeons for entire ingame days, and I believe the dungeon is small enough for characters with a decent memory to just still know where they're going.

In game solutions.

I am always a fan of these rather than meta-solutions.

The characters can come across another set of adventurers and they can either speak to them about or overhear them discussing the importance of mapping out underground systems or other areas that are tricky to navigate safely. Since you already bypassed a session 0 or didn't cover this it makes perfect sense to have this as a solution without you hitting them upside the head with the Meta-Game stick.

In my experience, and granted I have not played with inexperienced players for decades, I draw out or have the section of the map they are on visible with important things for me obfuscated obviously. When the characters move out of that area they no longer see nor have access to what I am trying to have them visualize. They don't get a copy of the drawing unless they create it themselves. Characters have to do the mapping as well and that takes time so depending on the detail it could take a few rounds or several minutes. This increases their chances of unexpected encounters.

Again, I play with experienced players and in some cases this allows me to be a bit more gritty with the game, but when I didn't a couple of times having them get lost usually gets the point across as well. At least one player comes up with the idea of mapping on their own at that point.

While it doesn't directly address your question, I think the top answer to this question might give you some valuable context to the place of mapping in D&D:

What's the point of long, empty hallways in dungeons?

Back in the day, there was this whole logistics minigame attached to light and mapping in dungeons. Since the minigame is long gone, the reasons for detailed mapping have changed.

Many of today's dungeons are actually fairly simple once you untwist the twisty passages (all alike). They mostly consist of a single primary path with a few dead-end side rooms, a few alternative paths, and a handful of points-of-interest. Why would the players need a detailed map in a dungeon with only four major room-complexes in it? Even if there are ten or more points of interest, if they're basically linear, why do you need charts to show you how to get around?

If you're putting your players into a labyrinth, and mapping the place is a minigame challenge of its own, that's fine; be upfront that they'll need to make their own map in there because the maze is part of the challenge.

But for your basic Phandelver or whatnot? Enforcing mapping seems boring and punitive, at least to me -- paperwork you're forced to do in order to get to the bit of the game you actually like.

Imagine playing some modern dungeon-crawler video game without an automap.

Imagine pausing your game every thirty seconds to grab your pencil and graph paper and make a map of your own.

There's a reason we stopped making games that way back around 1992.

Caveats Due to scheduling and attendance problems caused by being adults, my sessions are almost always three-hours-and-an-ending affairs. In these games, I need to keep the pace up and make sure we get a complete, satisfactory adventure; there's not much time for mapping or massive, complicated dungeons. I heard about Tremaux's Algorithm through one of my friends who runs a mega-dungeon crawl for his wife and kids. For them, it solves the problem of being able to ensure exploration of the whole space and finding their way back out again. What follows below are my pontifications based on this second hand information.

Analysis of the Problem

Your players will be exploring a complicated space and you don't want to just hand-waive navigating it. You want to bring the tension that comes from the possibility of getting lost in an unfamiliar space can bring, but you're concerned that the most obvious solution (drawing maps) might be overapplied and slow-down, detract from, or otherwise interfere with the game.

Alternative Solutions to Maps

If navigation and complete exploration of the space are your primary concerns, consider teaching your players about Tremaux's Algorithm for solving mazes. With nothing more than some chalk lines, they can ensure that each spaces is traversed twice at most. Here's how the algorithm works in theory:

1. When first entering a junction (this includes the first junction in the dungeon/maze), draw one mark in the hall from which you just exited.


2. From the hallway choices before you, choose (randomly or not) from among the hallways with the fewest markings on them (you should never have to choose a hallway with 2 markings). After making your choice, mark the entrance of that hallway with 1 mark of chalk and proceed down it.


3. Loop back to Step 1.

When you want out of the dungeon, just follow the hallways with only 1 mark in them.

Wikipedia link: Wikipedia Entry for Tremaux's Maze Solving Algorithm

With Tremaux's Algorithm, it's guaranteed that you will find your goal in the maze or explore the entire space, even if the space isn't a perfect maze.

Implementation: It will require some paperwork, either on your part (for reasons explained below) or on the part of the players, who can either make a simplified diagram of branches taken and not taken, or just keep a list of rooms and chalk markings make therein.

Disadvantages: As Slagmoth pointed out, any intelligent creatures will see the chalk marks on the walls and know what's up; they'll be able to set ambushes, erase marks, and draw new marks to trap you. There is also a concern in caves with high moisture content that the chalk might not stay. Also, someone is still doing paperwork to keep track of the marks.

Maps

Chiefly what a map does for you is show the position of locations relative to each other. Keeping track of the location of a particular chamber is easier with a map than with Tremaux's Algorithm. It also doesn't leave traces in the dungeon and the player's can take it with them when they leave. Mechanically speaking, the only real difference then is that your players are paying a few silvers for parchment, ink, and quills vs a few coppers for chalk.

Bringing It Together

Whatever your players choose to do, they now have a litmus test. Is it worth the mechanical component cost of creating the maps to create a tool to navigate the space.