Etydhv

In Avengers: Endgame, the following is part of the dialogue between Bruce Banner (BB) in astral form [who traveled back in time] and the Ancient One (AO):

AO: I can't risk this reality on a promise. It's the duty of the Sorcerer Supreme to protect the time stone.

BB: Then why the hell did Strange give it away?

AO: What did you say?

BB: Strange. He gave it away. He gave it to Thanos.

AO: Willingly?

BB: Yes.

AO: [long pause] Why?

BB: I have no idea. Maybe he made a mistake.

AO: [long pause] Or I did.

I'm unclear on what mistake she is referring to. I suppose it might just be her mistake might simply be not immediately giving Bruce the time stone, but that doesn't seem profound enough for this dialogue.

Also it's odd that she knows Dr. Strange's future, but doesn't know she gives Bruce Banner the time stone. Maybe this is relevant to her mistake (e.g., incorrectly perceiving the future).

Question: What is the Ancient One's mistake?

Her mistake was refusing to give Banner the Time stone.

Strange had to give it willingly to Thanos so she would be convinced that things really were that serious. Presumably in other futures he saw Thanos eventually gets it anyway, and then the others are unable to complete the time heist.

Question: What is the Ancient One's mistake?

The mistake the Ancient One was referring to was her initial intent to not give the Time Stone to Banner.

Before explaining, it's important to mention/keep in mind that the Ancient One has the ability to see the future (but only so far).

Going into the events of The Avengers (as shown in Avengers: Endgame), the Ancient One most likely foresaw Banner asking her for the Time Stone. However, as the OP quoted, it's the duty of the Sorcerer Supreme to protect the Time Stone at all costs. That being said, her initial intent was to not give the stone to Banner.

The reason why the Ancient One didn't already know about Strange's actions (i.e., Strange willingly giving the stone to Thanos) during the time of her and Banner's conversation is because she died in Dr. Strange; and, just as she told Strange right before she died, she wasn't able to see past that moment in time (nor was she able to see Strange's actual future... only its possibilities).

ANCIENT ONE: I've spent so many years peering through time, looking at this exact moment. But I can't see past it. I've prevented countless terrible futures. And after each one, there's always another. And they all lead here, but never further.

...

I never saw your future. Only its possibilities.

So, her coming to learn about Strange actions in Avengers: Infinity War was new information to her, and, with her knowing that Strange is the best Master of the Mystic Arts, she then says that perhaps she made a mistake [in judgement] when initially not wanting to give the Time Stone to Banner (and then immediately gives it to him).

The Ancient One is referring to her taking Stephen Strange as a student. She states that it is the Sorcerer Supreme's duty to protect the Time stone. Since Strange apparently willingly gave up the stone to Thanos she is worried that she chose wrongly in her successor as Sorcerer Supreme.

The dialogue of the scene continues thusly and further explains her fear:

Ancient One: Strange was meant to be the best of us.

Bruce: So he must've done it for a reason.

Ancient One: I fear you might be right.

The Ancient One believed that taking Stephen Strange as her student was the right decision. She believed he would become the best of them. Bruce Banner has just told her that Strange apparently shirked his most important duty. She is now questioning whether she is making a mistake in choosing Strange.