Innovation is a pretty big ask.
For most people! You came to the right place, motherfucker!
First, here's my cat naming algorithm. First, take a position. (No, not that position; nobody needs to see that.) What I mean is, take a title, like something military or royal. For example: Emperor. Then, an adjective of any sort, whatever pops into your head. How about Filthy? Then, name an article of clothing. We're gonna go with Mittens.
Boom. Emperor Filthy Mittens.
Again? Boom. Captain Saucy Pants.
Once more? Boom. Brigadier General Mousey Boots.
Sometimes it doesn't work. Rear Admiral Inappropriate Skinny Jeans is a little weak. (Or is it?)
Let's try to find a bad one. How about...Prince Regent Ugly Slacks. Or...Corporal Shitty Hat. Or...Sargent Fluffy Socks. Shit, that was really good. Is this infallible? Probably.
Epihipparch Racially-Motivated Sunhat. There we go. That took a while.
Now, for dog naming, I like to go someplace a little more personal. A favorite movie is a good starting point, and a great character within. Ripley from Aliens is a great example. Other great examples - Madmartigan, The Kurgan, Lo-Pan, The Dread Pirate Roberts, etc.
Sometimes, but rarely, you can get away with punny or cute, but I wouldn't go that way. The biggest stretch I'd make on this front would be something like The Dread Pirate Robert, singular. That's cute, but still encompassed by the awesomeness of naming your dog after The Princess Bride, and also naming your dog a pirate.
Things to avoid:
Don't name your dog Sit or Stay or any of that bullshit.
Don't name your cat with a soggy woman's name like Felicia or Priscilla.
Don't give your dog a human name that isn't funny. Doug and Gary are fine. Ken is not. (Though Kenneth comes around back to funny, if you're determined.)
Don't call your cat a traditional dog name. Fido the cat is only funny to you, and you're a fat loser.
Go easy on the names that 'mean something to you' but nobody else knows what the fuck you're talking about. Everyone knows the lady from Evanescence could sing, but that band sucked. Evanescence the dog is an assumed shit heel, and he'll get ridiculed in the park. (Note: Being known as a shit heel is the equivalent of being known as a cunt in human terms.)
Famous characters from literature and even history are almost always winners, as well as mythological names. This works for cats or dogs. Nero, Frodo, Ulysses, Earhart, Newton, Lucifer, Watson and so on.
Short Answer: These are my dogs, Ken and Adam. Hear it? It's bad. These are my dogs, Gary and Doug. Ding-ding-ding! Winner.