I know that a few in our sewing group make the dog beds for the local shelter and rescue organizations. They sew what I would call channels and stuff them full of scraps of fabric. That is where my leftover tiny scraps go.
I had a Westie and I just made a simple quilt using 4.5 strips, put a good quality batting in them. I still have the quilt top laying on my wicker bench in my bedroom to remind me of my beloved pet. Fabric looks faded from the washing but that is all, still intact.