What did Miss Maudie do that won Scout over in "To Kill a Mockingbird"?
I believe you are referencing what happens in Ch. 24. This is where Scout is trying to fit in to her aunt's world of dresses and social decorum. Alexandra is having several of her friends over to eat and gossip.
At one point, one of the ladies mentions how she won't hold her servant girl accountable for how surly and sullen she seems after the Robinson verdict.
This angers Miss Maudie because it represents the hypocrisy in Maycomb, the same type of hypocrisy that lead to Atticus being one of the few people who is willing to stand up and do what is right. In Miss Maudie's mind, the servant girl has every right to be sad and angry: an innocent man was convicted of rape. And all of these supposedly powerful and upstanding Christian women (and men) never lifted a finger to help out.
Scout realizes how mad Miss Maudie is, and she starts to get a little window into the hypocritical and racist side of Maycomb's citizens.
It is in this chapter that we learn of Robinson's death. Here Alexandra begins to wonder how Atticus will handle this and if it will be too much for him. She begins to wonder when the citizens of Maycomb will just leave Atticus be.
Miss Maudie reminds her that what Atticus has done is noble. Many people in Maycomb trust him to do what is right, that is to defend the African Americans.
This helps show Scout how loyal Miss Maudie is to Atticus and how much she admires him.