Role and Importance:

Mrs Dubose is a character in the novel which shows moral courage.
Despite her morphine addiction, she wanted to die 'beholden to nobody' even though she knew she would have to go through a lot a suffering without the help from the drug. Majority would have chosen the path of a peaceful death, but Mrs Dubose was determined to overcome her addiction and to die with the pain instead.
Her moral courage had also served as a lesson for Jem about real courage. Atticus had sent Jem to read to Mrs Dubose daily to show him what real courage was. He did not want Jem to think that 'courage is a man with a gun in his hand'

She's Got Moral Courage!

Mrs Dubose holds strong to her beliefs and goals, for example, she wanted to die as a non-morphine addict. And in the end, she managed to achieve her goal despite having to go through the pain without the help from the morphine, and also the temptation to die painlessly.

-"She said she was going to leave this world beholden to nothing and nobody. Jem, when you're sick as she was, it's all right to take anything to make it easier, but it wasn't alright for her. She said she meant to break herself of it before she died and that's what she did."
(pg.123 - Atticus)