About the time I was a high-school senior, I helped a family catch their cat. I asked them what his name was. They said they had a dog named Rusty, so when they got a cat they named him Corroded.
When we were kids a few years before and our cat had kittens, one was solid gray, and we named her Dusty.
What distinctive names have others given their pets–cats, dogs, whatever? 
I had a very small dog - chihuahua cross. I was really into mathematics. I called her Epsilon-pi. Epsilon in mathematics is a ‘very small amount’ and she was a very small amount of dog. Pi is my favourite number.
Everyday, she was just Epsi.
When I was a kid, we had a cat. I named him ‘Pud’ (rhymes with ‘wood’), because that was the sound his front paws made hitting the floor as he jumped down from something.
One night I was very tired and accidentally called my dog, Grady, “Gravy.” It cracked me up, and I told my parents and it cracked them up, so now we all call him Gravy.
Someone named their dog or cat “Murder”. I can imagine them calling their animal…
Murder!
Murder!
Murder!
I hope the neighbors never heard them.
When I was very young, we had a doberman called “Sprikey”. This was short for Sprechen Sie Deutsche.
I owned an Australian shepherd that I named Shadrach, and at the same time period an orange cat I named Meshach. Very biblical, but really just named after the old Brook Benton song.
I mentioned elsewhere that I named a cat Mousy Tongue and called him Mao for short.
Our last cat was Viralata (literally, ‘can turner’), which is Brazilian slang for a street person. I was studying Portuguese at the time. We just called her ‘Vira’ (like Vera with a long ‘e’).
Our newest inherited cat is just Kitty, but spelled Q’itih. She’s a tuxedo and looks like she’s wearing a chador, hence the Arabic-looking spelling. Drives the vet nuts, which was my intention.
I knew a woman whose cat was named ‘Battery’.
We got a poodle when I was in middle school. Because she had papers, she needed a regal name, because she was a poodle, she needed a French name, and because we are of Polish ancestry, she needed a Polish name. Ergo: Contessa Yvette Dombrowski, AKA, Cyd. 
My grandparents had a German shepherd who shadowed my grandfather, so they called him Chadeau - can’t just have the common spelling.
My youngest sister has a small zoo - her two cocker spaniels are Chivas and Bacardi, her greyhound is Otto, and her cats are Ludlow and Niblet. All of them have middle names because my sister is insane.
I had a parakeet named Chrysler. His name was the first thing he learned to say.
Thanks, Chefguy. I have been looking for that song for years… I’ll check it out on iMesh.
My wife and I, on a long bus trip somewhere in South America, decided that Olive and Violet were both very nice names. But the letter Q is sadly underutilized in English. So we would replace the O with a Q in each name, making them Qlive and Viqlet. Pronounced Clive and Viclet. She then decided that when we have a pair of baby goats on our farm, they will be named Qlive and Viqlet. Which never came to fruition, but the names are on hold, if ever needed.
Any dopers are welcome to use them, no credit required. Long South American bus trips are very nurturing to the imagination.
My current cats name is Waffles. I call him Bobbies.
I went from saying, “Hey my baby.” to “Hey my babies.” to "Hey my babbies.* and now it’s “Hey my Bobbies.”
*After “How is babby formed?”
Growing up, we had a neighbor named Professor Sullivan. He was the head of the drama department and, fittingly enough, he named his basset hound Gilbert.
I also had friends who owned an enormous, rather scary looking mastiff. He was a gentle giant all the way, but people would look at him and only see a big, frightening, dog. In a stroke of genius, they named him Willie Bite, because, of course, that’s the first thing anyone says when they see a large scary dog. Willie, recognizing his name and willing to make a new friend at any moment, would wag and smile, and show just how friendly he really was.
Conversely …
Some very gay neighbors had an adorable tiny Yorkie they named “Babs”. Babs was very enthusiastically friendly to all, wagging her whole body like mad the whole time. Between her Furby face and Yoda ears you couldn’t help but smile to look at her. She with her rhinestone encrusted collar and leash was the perfect fashion accessory to these two rather flamboyant (and very nice) guys.
If in our approaching dotage my wife and I ever get a dog, it’ll be a silly little yorkie like Babs. Named “Fëarsôme BeËste” in the style of Mötley Crüe - Wikipedia
At a meet up in the park Easter Sunday, I met a sweet Yorkie called Swiffer. She totally looked the part.
One of Dusty’s brothers was a black-and-white tabby with six toes on one front foot and seven on the other. We named him Ripley. Poor little guy. He was only a few months old when my sister, crying, came in from the back yard where she had found Ripley–dead. 
We got adopted by a stray kitty a few years back. Ivylad strongly prefers dogs to cats, so at first we were going to take him to the no-kill shelter.
However, the kitty, who knew who he had to convince, jumped up on Ivylad’s lap that first night, curled up, and purred for three hours straight.
We named him Rary. It’s short for Temporary.
My first cat was named Princess Alexandra of Denmark - Sandy for short. We have also had a cat named Bunwarrior (who ate a lot) and one named Cleocatra (a tabby with very distinct “eyeliner” markings).