I like picking common words or names! My obsession with that started on Neopets, the novelty of seeing a pet named just 'Jessica' opposed to 'x_Jessica_x_124643' feels like big bragging rights you know! (And it just looks way nicer too) Sometimes I really like fandom names too, or if the petsite is super popular and I cant get my hands on any names I want, I'll just mispell a name or word I like. I think in general I like really cute names, funny ones, or edgy ones.
 
SO. It depends on the site. But usually its FULLY vibes.

The best example, is Xanje. Xanje is a collect-a-thon with no pet limit, so, its a lot. Most people don't bother to name all their pets.
I try to name EVERY pet I get on Xanje (that I plan to keep) which is a very time consuming process. At this point, I've started using the wikipedia random button for ideas. But also just whatever the first thing that comes to mind is XD
 
I like to think of the names of virtual pets because they carry the identity of my game experience. While researching, I have seen that many players use creative names to increase attachment to their pets. The method I use is to choose names related to the pet's personality or the theme of the game instead of random picks. I also like to write a list of names before assigning to ensure consistency and gameplay vibe. This helps me create a bond with the pet and bring motivation to continue playing. Names have a great impact on the immersion of the digital world.