I have three pets that are just mine, a bearded dragon named His Holiness, a royal python named Svafnir, and a leopard gecko named Kami.
I didn't name His Holiness, but when I was still with my ex, we wanted to give him a nickname, since "His Holiness" is a bit of a mouthful to say all the time, so we decided to use the name of the Pope at the time, Benedict XVI, settling on Benny.
Svafnir is named after one of the eight dragons or serpents mentioned in stanza 34 of Grímnismál (the others being Góinn, Móinn, Grafvitnis, Grábakr, Grafvölluðr, Ófnir and Níðhöggr). According to some scholars, "Sváfnir" means "the one who puts to sleep" in the sense of "death", which for a python was a pretty good description.
Kami, and his brother Kyu (who sadly died last year), underwent a pretty convoluted naming process, starting with 守宮, yamori, the Japanese word for gecko. I took the kun'yomi reading of 守, kami, and wrote it as 神, which also happens to have a kun'yomi reading of kami, meaning "spirit" or "god" (continuing the theme set by His Holiness and Svafnir). I then took the kan'on reading of 宮, kyū, meaning "palace" or "sign (of the zodiac). It's also the kan'on reading of 旧, meaning something like "old things". He was quite small, and potentially ill when I got the two of them, and I was hoping he would live for a long time with the proper care (sadly didn't pan out).
I also named my mum's cat, Odie, after the dog from Garfield, in a similar fashion to ironic names like "Tiny" for someone quite big. He also had a sister (unfortunately run over two years ago) that we settled on calling Mina. Originally she was called Bella after the main character of the Twilight books, but somehow we ended up with her (I forget how), and since none of us like Twilight even a little bit, we decided to name her after a female character in another vampire book, and what's more well known the "Dracula" in that sphere.
As for future pets, I've got Gilly for a dog, Jörmungandur ("Jörmi" for short) and Sith for cats, Moruk, Vishap, Kur, Tiamat, Grom and Tora for bearded dragons, Ofnir, Typhon, Skog, Bashmu and Lotan for snakes, and Tora, Tama and Kago for geckos: