Sheila, Sorry to hear about your Mother.
I think your idea to make a quilt out of her shirts is a good one. However, this sounds easier than it is. I would think a "rag-type" quilt would be out, because the sweatshirt material won't "rag"...at least, I don't think it will. So your choices will become more limited.
I would suggest cutting your shirts to the size you will need, then applying a light-weight fusible backing on each square. I don't think I would add batting, unless you just wanted to add a flannel sheet or something really light. Sew your squares together, add the backing, and quilt as you would any other. Bind it and you're done and you have a beautiful reminder of your Mother!
Good luck.