Mrs. Dubose is courageous because she is trying to kick a morphine habit before she dies. At the old age of 90 - 100, there must be considerable pain in her body. Morphine would play a role of at the very least lessening that pain. We are never really told what ails her so, but know that her experience of having the kids read to her helps make the distance between morphine shots become longer and longer until they just don't happen anymore.
Atticus cites her courage in these words:
“She had her own views about things, a lot different from mine, maybe… son, I told you that if you hadn’t lost your head I’d have made you go read to her. I wanted you to see something about her—I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It’s when you know you’re licked before you begin but you begin anyway and you see it through no matter what. 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.
Another way that Mrs. Dubose's situation exhibited courage is the fact that she knew she was going to die and soon. She knew her life was over. She could have chosen to die comfortably, but she chose to die free. That is admirable.
To add to Missy575's response, Mrs. Dubose has the courage to teach the kids a lesson in respect and dignity.