The name can definitely be considered a symbol in the story. In "Everyday Use," Dee decides to change her name to Wangero. We learn that Dee believes that using an English name is like being named after her oppressors (as somewhere in her genealogical tree, there probably was a black woman who was renamed by her owner and denied her identity). Therefore, the name signifies her liberation, and her rebelliousness. It also shows a return to her African roots.
For the mother and sister, the name is a symbol of the distance that is found between Dee and the rest of her family. It is obvious that they do not share her views on politics, identity and oppression. Instead, the women most likely believe that the fact that Dee rejects her name implies a rejection of them and everything else that her family represents.