I tend to pick my mabari's name based on my character's.
My first dog cooresponded to Thanatos Cousland. I didn't really have a cooresponding name, so I named him "Akita." (My brother hates that breed.)
After that it was Shiva Surana, and when I realized he got to have a dog I alt-tabbed and looked up Shiva from Hindu mythology. He turned out to have a pet bull named Nandi, so the dog's name was easy.
Ares Brosca's dog was named Mars, after the Roman name for the Greek God Ares.
Umbra (latin for "shadow") Aeducan is the proud owner of Aurum (latin for gold, literally means "shining dawn" iirc.) In the same vein Midori (Japanese for "green") Mahariel named his "Akatsume" (Japanese for "red claw.")
Jeeves Tabris was actually named for a Jerry Seinfeld quote. "You know, when you name your son 'Jeeves', you've pretty much set up his life's path for him. What is he going to become... a hitman? 'Pardon me sir, but I'm afraid I must whack you." I decided that this name fit due to Jeeves' destiny as an expendable workerbee for rich humans, which he completely flipped off to become a terrifying Assassin/Ranger/Shadow who ruled both Amaranthine and the Denerim Alienage as a lord. The only dog name I could think of to fit this combination of ridiculous and badass was "FuLin," (or "lion dog") which refers to a stupid looking breed of tiny Chinese dog that nobles apparently threw into would-be assassins' faces. (The dogs were trained to hold on to this face with their teeth.)
Gawain Cousland was an easy one. Gawain of Arthurian mythology had a horse named "Gringolet."
Sarai Amell was also easy. Before the biblical figure Sarai changed her name to "Sarah," her husband's name was "Abram." (Her husband changed his to "Abraham.")
Diana Mahariel's dog is named "Aktaion" because I am a bad person.