"Everyday" is an adverb that tells when. "Use" is a verb (with object) which means "to put into service;" a verb (without an object) which means "to be found;" and a noun which means "the act of putting into service."
So, what is the object? It is the "quilt." Mrs. Johnson wants the quilt to be put into service everyday by Maggie. She does not want the quilt put on the wall and not put into service every day.
The object is also "culture." Mrs. Johnson wants her cultural legacy to be put into service, not used as a passing fad.
"Service" has a utilitarian function, which means "having regard to utility or usefulness rather than beauty, ornamentation." The quilt and the African American culture is something to be used, to live. I think it's the verb, not the noun. It must be acted on everyday, not ornamental.
everyday use refers to the quilt. she doesn't want to give it to wangero because she is going to put it on her wall as a decoratioal art peace as for Maggie she wanted to be the quilt that she made her bed with after she got married as tradition and becuase they were poor. Mama new that Maggie would appreciate the quilt more because it was made from the hands of her ansestors and not treat it as if it was an artifact.