I rename all my mounts and pets, sometimes it's just a name which comes into my head when I get them, sometimes I go looking for a meaningful name and sometimes it's a reference to something else.
For example I called the elk mount Halla after the deer from Dragon Age, and the black horse Liadri after a Guild Wars 2 character. My sabre cat is called Gwenodd which is Welsh for 'smiled' after the species name for the sabre tooth tiger - Smilodon populator. (Possibly the most dramatic species name ever, it roughly translates as 'knife-tooth the destroyer'.) But then the shadowghost horse is called Brith, which means speckled, and the white horse is Luna, as in the moon.
Pets tend to have sillier or less formal names. I called the vermillion scuttler Fluffy because he isn't and the Senchal Striped Cat Gobbolino after a character from a series of children's books. The beta monkey I called Simach which is literally just monkey in Welsh (after calling it 'my monkey' for ages before finding out I could rename it anything else seemed weird). Oh and after I called Sanguine's Black Goat Anakin for his ambiguous morality and annoying bleat I had to call the Ninendava Sacred Goat Padme.