Atticus feels that Mrs. Dubose is a great lady because she is strong-willed and proud, and lives life on her own terms.
Mrs. Henry Lafayette Dubose is the neighbor whose house is two doors to the north (Ch. 1). She is described as “plain hell” and “the meanest old woman who ever lived” (Ch. 4) and is a fixture in Scout’s childhood. For most of the time Scout is growing up, Mrs. Dubose is just there, yelling insults at the children as they pass by even if they attempt to be nice to her.
Atticus is always polite to Mrs. Dubose, despite her rudeness. He is polite to everyone. The encounter Jem has with Mrs. Dubose’s flowers gives Atticus an opportunity to teach his children responsibility, but after her death he also lets them know why he considers her such a courageous person. After Jem destroys her flowers, Atticus has him go to read to her. He later explains that he wanted his children to see what real courage is. Sometimes courage is mental as well as physical.
Mrs. Dubose was addicted to painkillers, and she wanted to wean herself off of them before she died. She was having Jem read to her so that she could have a distraction. Jem did not know any of this until later, but Atticus knew that he could explain it and Jem would understand.
You rarely win, but sometimes you do. Mrs. Dubose won, all ninety-eight pounds of her. According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew. (Ch. 11)
Mrs. Dubose was an underdog. She was addicted to morphine, but she kicked the habit. What she did took immense courage. Atticus wanted his children to see that sometimes you can win an unwinnable fight, and sometimes just trying to win when it seems impossible is courageous. Atticus himself was facing an uphill battle that would require courage of him and his children as the trial began. Showing them Mrs. Dubose was a way of sending a message about different types of courage, and courage being found in the least expected places, to encourage them to find it in themselves.