What does Atticus do that causes Scout to say she felt he was the "bravest man who ever lived"?
When the children accompany Atticus on his walk home from his office, they witness Atticus greeting Mrs. Dubose with chivalric gestures, Scout finds him "the bravest man who ever lived."
Throughout this bildungsroman, Scout and Jem learn much about good character and about biases. Scout certainly learns that bravery is not defined by aggressive behavior. Bravery is resisting primal, irrational urges and using intelligence and understanding to assist in determining one's actions, instead; bravery also, involves not taking advantage of someone who is weaker. This is why Atticus stopped hunting; he felt that with his eye/hand skill, he possesses an unfair advantage over prey animals.
These examples of bravery are part of the thematic message of Harper Lee that resisting the status quo is as brave, at times braver,than aggressive behavior. It is this same bravery that leads Atticus to accept the position of defender of Tom Robinson in his legal case.