What do you believe it means to be "weary," physically and mentally, In an era before the American Civil Rights movement, what social and economic conditions might have contributed to the sense of weariness expressed by the musician?
Truthfully, the only way to know why Langston Hughes titled his famous poem "The Weary Blues" would be to ask him. Given that we no longer have that option, your opinion is as good as mine. However, as you noted in the question, there might have been good social reasons to include the word "weary" in this particular poem. The writers of the Harlem Renaissance, of which Hughes was a part, had good reason to be weary. They were fighting an uphill battle, both as Americans of color and members of a cultural movement to be taken seriously as practitioners of their common craft. Hughes' emphasis on weary also applies well to a blues musician, a piano player working in a bar for a living had to play long hours for people who often did not appreciate that fact. One would indeed be weary by the end of the night.
But I don't think Hughes was only being literal. I think his use of the term "weary" is specifically meant to point out the body and soul weariness of trying to be accepted as equals after so many generations of slavery and maltreatment. Plus, playing for whites, as so many of the jazz musicians of the time did, would add another element of weariness to people who were already carrying a heavy burden.