I have a cat named Don Quijote because when my roommate found him he was just skin and bones. We couldn't keep the cat at the house because of the landlord, but I took him to my mom's house and she thought of the name.
I named my bullfrog Finduilas after an Elven princess.
My mouse didn't have an official name, but I called him Lembas sometimes because he was small and reminded me of Elvish waybread.