To add to the good points made by the previous poster, I'd like to suggest a couple of specific themes:
1. The duties of the black artist: Hughes' work is full of questions about what the African American artist should or should not have to do. See, for example, his famous essay "The Negro Artist and the Racial Mountain." Walker's story "Everyday Use" has a black artist in it, too -- Dee, who in many ways is the fictional counterpart to Walker herself. Most readers don't like her, viewing her as betraying her true heritage for something make-believe. I think that Hughes would support her, though. Why should she have to stay down South? Stick to her given name and religion? And so on.
2. Similar to #1, both Hughes and Walker point to the conficts and tensions within black identity. In Walker's story, for example, which is more "black": Christianity or Islam? Life in the urban North or life in the rural South? etc. Hughes' poem "Theme for English B," along with many of his other works, show similar conflicts or tensions.
It's worth noting, of course, that the two writers are from very different periods. To be more precise, Alice Walker is very far removed from the Harlem Renaissance. The dates for the Harlem Renaissance vary a little, but it's generally said to have ended by the mid-1930s. Walker's story "Everyday Use" is published about 40 years after that, and this work is much more closely connected to what's called the Black Arts Era (or Black Arts Movement) than it is to the Harlem Renaissance.
Most definitely Langston Hughes and Alice Walker share common themes. Consider all of their works - they are Black Americans, writing about exactly that. Although Alice Walker wasn't technically part of the Harlem Renaissance, she references the same themes, sentiments and goals in her works that the Harlem Renaissance started.
That said, "Everyday Use" is a short story that draws from themes of heritage, overcoming oppression, and hope and strength from unlikely sources. The fact that the story is told from the perspective of an African American and a woman is further evidence that Walker intends to heighten the inner strength she herself has found as a result of where she has come from.
Langston Hughes (and other HR writers) did the same thing. He wrote from his heart and soul to show that although America was not ready to fully accept it, his heritage had much to offer in the way of inner strength and overcoming oppresssion. These themes were manifested in his art. This is essentially the overriding theme of the entire Harlem Renaissance. Black America found its voice - in a very unlikely place - and virtually changed history as a result.
Alice Walker has continued in this spirit - along with many of her black female writer counterparts - like Maya Angelou.