It is not Jem, but his father, Atticus, who states in "To Kill a Mockingbird" that Mrs. Dubose is the most courageous person he ever knew. Atticus states this because he admires Mrs. Dubose for displaying courage by fighting to give up morphine even though she knew she was going to die.
In "To Kill a Mockingbird" by Harper Lee, Atticus Finch makes his son read to Mrs. Dubose every day. The reading sessions are punishment for tearing up Mrs. Dubose's flowers after she insults Jem's father. At first, the sessions are cut short by Dubose's irritability, but gradually last longer and longer. A short while after the punishment ends, Jem learns that Mrs. Dubose has died.
Jem receives a single white flower from Mrs. Dubose as an inheritance. Atticus explains to Jem that Mrs. Dubose was dying from cancer and hooked on morphine for the pain. He calls her courageous because she fought to kick the habit and had a clear mind before dying. The reading sessions with Jem helped her do so. Atticus feels that it is courageous to keep fighting even after knowing that she had lost.