Almost as hard to come up with as names for your children. Although occasionally, it seems like people just don’t want to think about it too much. I can remember being a kid and having a cat named Lucky, and then the next cat we got was also named Lucky. Ironically neither cat was.
We had a lab cross named Penny, and a beagle named Tammy. Both names that were considered for me apparently, but were given to dogs instead.
I can remember sitting around the kitchen table writing names on paper that would be chosen out of a bowl. That was for our black shepherd. I wrote Thor, a brother wrote Brutus, and he ended up being called Ben. That was a good name for him.
I had guinea pigs named Bubbles and Pickles, I have no idea why. I was a kid.
Since I got married and started my own family we have had a cat named Dickens, not after the writer, but because he was a dickens. Also a cat named Claude, because that was what you got when you went near him “Claw’d”. Our kids names their cat Skittles (why?) and our first dog a golden retriever was a purebred and so appropriately we named him Montgomery (Monty). Two rabbits, Magic (black) and Twister (because he would run and then jump up in the air and twist around, also the movie was a big hit at the time) Our current dog is Tira, named by our kids after some anime character.
I always think of good names, like Achilles, so I could say “Achilles heal! When we don’t have a pet to name, but when we do have a new one, it just seems so daunting to come up with the right one.