MSQ recently had a tutorial to make pet beds from fleece. My "problem" with that is now to get fur off of/out of fleece but anyhow....
Check out your local thrift stores for pillow shams for availability and price. Your bag is already made for putting scraps in, whip the back shut and the pet bed is ready to go for your own fur baby or donation to the local animal shelter. By the way, they can use donations of plastic bags that stores provide for your purchases, too! A good way to recycle those.
Have you seen the pet beds made with discarded long-sleeved pull-over shirts? Those are cute! Sew the neck shut, fill the arms and body with your scraps and close all open ends. Then pull the arms around the body and stitch those to the body to make a cozy bed for pets.