Just what sounds "right". My daughter named her cat "Comet" because I once said I wanted to name a cat that...it followed in series of my other cats I had when they were small, Ajax and Babo ... both were names of bathroom cleaners.... However, I call her cat "Fred" because that is what seemed to fit (and annoy her...cause you have to annoy your 13 year old daughters). However, she calls him "The Rat" when she is upset and he answers best to "Kitty" for her. He answers to "Fred" for me. "Cat!" is also a name name that works.
The dog is "Missy". She was such a scared skinny thing from the pound and it matched her well. The name she had before was "Ladybug" but it didn't fit. She is my son's little "Missy". She can't wait for him to get home each day and is devoted to him...though she loves the rest of us too.
We had a Sunfish fish named "Sport" because he loved to come to the side and follow my husband the room from his aquarium. He also would beg for worms and other treats my husband brought him. Hoover was the algae eater because he vacuumed the tank. I can't remember the other fishes names but they matched their personalities...(we had two bluegills that worked like an evil gang, one would chase the feeder fish out of hiding and the other would wait for dinner to arrive - and a depressed mud catfish that would jump out of the tank if we didn't keep the lid on (he would dry out but would come back to life if you put him back in the tank)...we were going to return him to his pond but my brother-in-law found him on the carpet and threw him in the dumpster because we forgot to tell him to put him back in the aquarium even if he looked dead.
Prior to that, our dogs always had names that were tested by my husband calling for them loudly. If you couldn't call them from the woods easily, the name didn't get chosen. Our redbone hounds were first a male named Red and then the female was named Pepper. We later had Peaches. And my kids named the next dog "Hank" after their favorite book series "Hank, the cowdog".