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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).

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.

While "not immediately giving the stone to Banner" is a sensible answer, I agree with the OP's original feeling that this does not seem to carry the amount of gravitas this exchange has. So I propose an alternative. She made a mistake with the dogmatic assertion of protecting the time stone at all costs, and moreover she failed to even execute that properly if it was even possible for there to be a future where the best possible option available to the best possible Sorcerer Supreme is to give up the time stone and rely on a complicated time travel plot to fix things.

The rigid dogma would have constrained original, out-of-the-box thinking, which are the hallmarks of a successful Sorcerer Supreme. It may not be clear from the movies, but they are often tasked with warding off threats of vastly superior strength, and clever exploits like with Dormammu, who was radically stronger than Strange, are necessary. And with better planning and training, maybe a situation like this could have been averted in the first place. So her successor inherited a bad, inflexible situation from her and had to take drastic measures to have even a slight chance to fix it as a result, and now she must trust Dr. Strange's unknown plan, and to rely on a time traveling Hulk to succeed both in his reality and in returning the stone.

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.

Marvel seems to have established some form of "many branching worlds" time travel rules for its universe. No "Back to the Future" rules, where time travel to the past can rewrite your own timeline. Time-traveling to the past, and mucking around, can't change the Avenger's future, or undo the snap.

Banner and our Avengers come from a timeline where The Ancient One did not give the Stone to Banner - that is what she implied was her mistake.

The Ancient One that Banner met will now live in a timeline with no Thanos. The Avengers killed him in the final battle of Endgame. By giving Banner the Stone, she saved her reality/timeline from the Infinity War.

I believe the answer is as Legion600 suggested, that the Ancient One is considering whether it is a mistake to trust Stephen as the future Sorcerer Supreme. But I don't think that answer explains what is going on in the Ancient One's mind very well.

Giving the Stone to Thanos sounds, on the surface, like a terrible idea. The shock of that is making the Ancient One question whether the core underlying mistake was to trust Stephen with the Time Stone in the first place, if that's the sort of bad decision he would make.

But the dialogue continues:

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.

This shows the Ancient One coming back around. She considered whether she might have made a mistake about Stephen. Hence all the long pauses in the dialogue in the OP's question; she's thinking hard there. But she concludes in the end that she does trust Stephen, and believes he is "the best of us". Therefore Stephen must have taken the course of action that would have the best hope for a good outcome, strange as it may seem. Therefore, she should give Bruce the Stone, as that seems to be the path that Stephen was trying to steer towards.

That moment of doubt and then reconfirmation that she trusts Stephen is the crux of the scene. It's what breaks through to her to convince her to give away the Time Stone.