In the story "Sonny's Blues," by James Baldwin, I think we can take the title to mean more than one thing.
Sonny is a main character in the short story, and he and his family have had their share of suffering (a major theme). His father saw Sonny's uncle killed by a car full of drunken white men and hoped for better days but died never seeing them—still suffering; his mother worked long days raising her family in Harlem; Sonny's brother (the narrator) has lost his daughter Grace, and Sonny has been faced with drug addiction, jail time and feelings of abject isolation.
The narrator was charged by his mother before her death to care for Sonny, but the narrator isn't sure how to do this: he does not understand Sonny, his choices or his love of music. With this kind of family "history," Sonny's blues can refer to his feelings of sadness or depression, which we often call "the blues" or feeling "blue."
However, quite literally, Sonny's "blues" also refers to Sonny's love of music—of muscic called jazz and the blues. All that matters to Sonny comes in the form of music: it speaks to, and feeds, his soul like nothing else can. It transports him to another place, but that is not to say it is a place of joy. He tells his brother:
"Sometimes you'll do anything to play, even cut your mother's throat." He laughed and looked at me. "Or your brother's." Then he sobered. "Or your own."
Sonny exists in a lonely place where few people in the world really understand him except musicians. It is not until his brother comes to hear him play one evening at a night club that his brother realizes that Sonny has a gift for music—and while it soothes him, it may perhaps haunt him as well. The narrator notes:
...Sonny played. Every now and again one of them seemed to say, amen. Sonny's fingers filled the air with life, his life. But that life contained so many others... Then he began to make it his...it was no longer a lament...Freedom lurked around us and I understood, at last, that he could help us to be free if we would listen, that he would never be free until we did.
If nothing ever happens to really release Sonny from his internal suffering, we can infer that it is eased when his brother comes to understand the music and what it means to Sonny.