aside from going against your mother's wishes...finishing the quilt depends on the age of it.
if you do not have the same era of fabric, then it will look ODD when finished.
If it is really old/vintage, then the historians say DON'T finish it. But keep it as it with a label attached to it giving the history of it.
as for FRAMING a block...please don't. Cotton is considered a living fiber and once it is encased in a frame it will dry rot and be lost. Make your mom a small pillow out of it, or a wall hanging!