Whatever fits the personality/appearance of the pet. I usually go for serious names, though I've been known to throw the occasional silly name in there. My current stable consists of:
Forserious Names
Shia the (Loque) Spirit Beast...because Shia is such a quiet, wise-sounding name, and that's what she is.
Goliath the (King Krush) Devilsaur. Because he's a huge monster
Grimshaw the oil-coated Wolf, it's an old-fashioned name that I felt fit because of his black coat and the nickname "Grim"
Morgana the oil-coated Hawk. Duno, such a dark-sounding female name seemed to fit her appearance.
Aether the sunwell dragonhawk. Aether is "the ancient Greek personification of the clear upper air of the sky". And also the name of my blue d-hawk mount.
Dargonax the Chromaggus. Dargonax means "devourer" in Draconic, which easily fits a two-headed dragon-dog.
Myst the (temp? hopefully not) spirit Wolf. I just feel Myst suits a ghostly pet, it was once the name of my Gondria.
Ashmane the flaming boar. I was having trouble with thinking of non-silly names for him before I remembered the boars in Blasted Lands and realized how well that particular mob's name suited my boar as a personal name.
Forlol Names
Splat the white moth. Obviously a silly name, and it always made raid members chuckle when I'd say after she died, "OMG! Splat went splat!"
Joystick the ghostly jormungar. Silly name with light innuendo. Couldn't help myself >.>
Asunder the black outland raptor. I'm not very creative with reptile names, so I named him this because I was only taming him as... A Sunder. *badumching*
Precious the Princess with ribbon, a /nod to my favorite mob in all of Icecrown, even though I never did see her ribbon drop.