Scout is saying she wants to be a better lady than Stephanie as she is teasing Scout, and Miss Maudie is trying to comfort her and also keep her from saying something snotty.
Scout has a tendency to speak her mind. She also often gets in trouble for saying things to adults that she maybe shouldn’t have. However, in this conversation, the adults are doing something that adults often do but should not do—they are teasing her.
Look at the way the conversation between Stephanie and Scout goes.
Miss Stephanie Crawford called from across the room, "Whatcha going to be when you grow up, Jean Louise? A lawyer?"
"Nome, I hadn't thought about it..." I answered, grateful that Miss Stephanie was kind enough to change the subject. Hurriedly I began choosing my vocation. Nurse? Aviator? "Well..."
"Why shoot, I thought you wanted to be a lawyer, you've already commenced going to court." (Ch. 24)
When Stephanie Crawford asks her what she wants to be when she grows up, the age-old question every little kid gets asked, Scout thinks it is a legitimate question and tries to quickly think of a legitimate answer. However, Stephanie is just using the question as a way to tease Scout about watching her father during the trial.
Stephanie is just being plain mean here. They continue to tease Scout about her clothes, for example. Miss Maudie, who is a genuinely nice woman and an actual friend of Scout, who doesn’t pick on her and treats her with respect as if she was an adult, is trying to help her. She holds on to her hand in a show of respect and support.
Miss Maudie's hand closed tightly on mine, and I said nothing. Its warmth was enough. (Ch. 24)
Miss Maudie does not need to say anything. Scout knows that she has a friend there for her, even as the ladies are teasing her. When she says that she wants to be a lady, it is easy to interpret this as an insult to Stephanie, because she isn’t acting like one. This is why Stephanie has to stop and think if Scout means any “impertinence” (disrespect). The comment can be interpreted innocently, meaning Scout wants no profession. It can also mean she wants to be a better person than Stephanie.
Throughout the course of the trial, Scout is put through many tests of courage and character. Having a father that stands up for what he believes in has taught her that she has to do the same. However, she is often put in difficult situations, which is hard for someone so young. Scout is always independent. Her father’s decision has made her a target of much ridicule, and made her need to grow up fast.