What sort of "Everyday Use" do we figure Dee would put the quilts to if she were to be given them?
The idea of the story, "Everyday Use" is that Dee wouldn't put the quilts to everyday use. That's the difference between her and the other two members of her family.
When Dee lived with her family, apparently, the family was rural and backward and "beneath" her. Currently, she has become educated and hip and urban. What was once backward is now quaint. The quilts are something from her past, from the past of her people. She sees the quilts as works of art, not as things to be put to every day use. To Dee, using the quilts to, say, cover up on the couch or use as a comforter on a bed would be a travesty and a waste. She would, most likely, hang them on a wall as decorations or display them in some manner.
Dee is a "new" black woman, breaking away from the boundaries society placed upon black women for years.