We don't know a lot about Daddy's sister, Ellis, except that neither Daddy nor Billie Jo want Billie Jo living with her when she offers to take her in, and that she lives in Lubbock, Texas, not far from the panhandle. Evidently there was major sibling conflict when Daddy and Ellis were growing up, and he's concerned about how she would treat his daughter.
Based on what little we know, the original plan was for her to raise Franklin as her own child, perhaps because of the tragic circumstances of Ma's death and out of Daddy's desperation to give him a better life. When she arrives and learns of Franklin's death though, she responds coldly, refusing to hold the baby or even speak with Billie Jo. The fact that she only speaks briefly with Daddy, then "turned around / and headed back to Lubbock" (38.1) reveals a hardness and bitterness that hardly makes her fit for caring for children.