Cut bottom in to a square (or something that resembles a square)
Sew top and bottom leaving a space to turn and stuff. I sewed with contrasting thread so you could see. (not really, I just didn't bother to change the thread in the machine)
Turn right side out.
Look at the fun things you can use to stuff your pet bed. Holey, mismatched, or stretched out socks. Quilt stitch-outs. Left over batting that is too small. Left over pieces of fleece. Really, anything somewhat squishy -and washable- that you have laying around would work. *note: I did cut the socks up the middle so they would wash and dry better and not be too lumpy in the pillow.
Stuff your pillow and sew it shut. (yeah, not pretty, but functional!)
You could totally get some nice fleece or some printed cotton and it would work too. I think the brilliance is in the stuffing.