I too have struggled with names for my pets sometimes. It's very rare if I have a name ready to go at the time of taming.
I draw inspiration from a variety of sources.
Sometimes it has to do with an animal's attributes. For example, let's say I'm looking at taming a cat. What are its colors and/or patterns? What kind of personality do I envision this particular animal to have? Once I've locked onto an idea, I research names on baby and pet naming websites that have those qualities in their meaning. So let's say I'm thinking of the word 'graceful' for my cat. I pick a gender and scroll through names that mean 'graceful'. If I don't happen to find anything I've fallen in love with from that point, I'll look up related adjectives such as 'athletic', 'poised', 'agile', 'dancer', 'sure-footed', etc. If that method doesn't work out, I pick another attribute about the pet I like, such as the fur or eye color, and follow the steps all over again until something clicks.
On other occasions, I pull inspiratation from outside resources such as books, video games, RL pets, WoW friends, RL family, etc. Perhaps you have a favorite character from a book you've read or a game you've played that would suit your new companion, or maybe you want to pay tribute to a childhood pet you once had by having them take part in your adventures throughout Azeroth. If you find common names too boring, you can always put a spin on it by altering the spelling or combine it with another name to make it appear more whimsical and inventive. For example, I have a DK named Avariel which is a combination of the names Ava + Ariel. I also have a huntress named Lilyth which is just a variation of the name Lilith.
Maybe there's an overall theme you'd like to have between your hunter and their pets. Are you into astrology? Names of famous astrologers and constellations are the way to go! Are you a fan of music? Name all of your pets after musicians you admire! Do you like to travel? Use famous cities, destinations, and landmarks as naming ideas for your pets! The added benefit of this concept is that you have a nice array of names to pull from, not just for one pet but for any additional pets you may want to acquire for yourself down the line.
The most important thing to remember is that pet names aren't permanent in the game. You can change your mind at any time. I've changed my mind numerous times regarding what names I've given to my own pets. Over time and through many years of playing WoW, I had to teach myself to stop obsessing so much over what names to give my pets. And don't beat yourself up too much over not finding a name right away! It's happened to all of us more times than we care to admit. So long as the name has some measure of personal value, or if you just like the way the name looks and sounds, that's all that really matters in the end.
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