This happens in Chapter 11.
At the start of the chapter, Jem really hates Mrs. Dubose. She says such horrible things about Atticus that he cannot help but hate her.
But then when he ruins her flowers, Atticus makes him go and read to Mrs. Dubose every day. He still does not really like her, though.
Then she dies and Atticus says that she was a great lady because of the way she fought her morphine addiction. Because of this, Jem seems to be rethinking his attitude towards her.
He doesn't exactly come to like her, but at the end of the chapter he's toying with the flower she gave him and thinking. It's implied that he's thinking about her.
Jem idolizes his father, Atticus and only for his father's sake does he attempt to ignore the blatantly rude insults Mrs. Dubose yells out as he passes. No longer able to contain his anger, Jem retaliates ruthlessly by ruining the one thing Mrs. Dubose seems to have treasured: her flowers. In order to atone for his childish actions, Atticus forces Jem to read for the feared and supposedly dangerous Mrs. Dubose once every day. Scout, as a devoted caring sister agrees to accompany Jem in order to provide a source of courage and comfort to him. As many reading sessions pass, the two children are able to observe Mrs. Dubose in great detail and are able to conclude that she may have a disease of some sort. When the news of her death arrives, Atticus explains to the two that she was in actuality, a courageous and persistent old woman who had lived her disease-riddled life under the influences of a drug that eased her pain called morphine. For years, she has succumbed to her addiction of morphine but died with what remaining dignity she had: free of morphine. Free to die a peaceful death.
Blah blah blah blah blah. Yup; that's the answer. You're welcome!