I actually wait until I've got a whole ton of un-named, then I name as many as I can all at once using the powers of CHAOS.

Usually this means looking at them, and randomly choosing words that come to mind for them. It has produced some fun results:

PinkSpikyThingFromTheSea


FLOWAH


Turd


The Wyrd

Turd

Lol, Amarok. I was about to defend your poor Lupora, but yeah, she's well named. ;)

My naming process changes. If fact calling it a process might be a touch overkill.

Duke Cunning of Plan and Duchess Cunning of Plan are part of a joke from a tv show, Blackadder. They're two of my starters for a project.

Any project-related bred pet gets a terrible mix of letters and numbers. Possibly a little Borg-esque, but it helps my goldfish memory. Like the snappily named, VP2 UFMFWWMHFcar.

For pets rather than projects, I use a mix of God/Goddess names and fictional character names I like. I can forget I've used a name sometimes, however. I think I had three pets named Cillian at one point.

My battle and mission animals tend to go with a theme.

The lupes have a moon theme (obv.) - which can also incidentally be Gods and Goddesses
The flower themed have a poison name theme (which actually makes sense in hindsight - plants are full of horrific alkaloids.)
When I make my sylvorpa party, they must have a fractal theme: mandelbrot, julia, and one other, I'm not sure what yet.

Like most of the people on here, it varies for me. Most of my pretty pets are named randomly, just me coming up with stuff, most of my themed pets are named words in different languages based on their theme, and most pets involved in breeding projects are named for their traits.

For my themed pets, I usually just go to google translate, put in a word that has to do with their theme (for example, with my Arctic Griffies, I looked up words like 'cold' 'snow' 'storm' 'ice' or 'freeze') and start flipping through languages until I find something I like.

How do you guys even use these specific names? Or one certain place. I have many google translates, random name generators, "lets-look-at-it-and-make-a-nice-name-for-this-ugly/pretty-pet" is probably the most common though.

With me, it matters. Sometimes it's from how it looks, especially with Themed (like my Chocolate Delight Zolnixi had names that reminded me of something a Chocolate Bar would be named, while my Tiger Lilly Lupora are Tiaga and Lilli). Other times I reference various RPGs I have played in the past (I've got Fire Emblem Awakening Lighira and Aurleon, CIMA: The Enemy named Zolnixi, Wild ARMs 3 named Ryori, and Tales of Destiny 1&2 named Ny'vene). And sometimes it's in another language in whole or based on a word (I've got Black Cat Vulnyx named Kage and Umbra, and my first Aurleon was named Suzako).