I'm part of the "if it catches my eye" club, but I am very fond of double letters! I have a good chunk of my total pets that fit that description.
BIRD NAMES I'm picky, but I'm in love with them; I have Hawk, Jay, Kestrel, Kite, Rook, Talon, Avis (Latin for bird) and I've seen Skraye referenced as a name for a sea tern, though tbh it's just a name I made up a long time ago that stuck.
Some other themes I like:
Earthy/Universe: Ivy, Geo, Mica, Nimbus, Quasar, Rose, and Terra. Historical/religion/mythology: Cross, Cain, Halo, Haven, Sutra, Ceres, Paris, Loki, and Orion. Names of places: Paris, Surrey, Avis, Harlem, and Rio.
I'm especially happy when a name overlaps two themes that interest me or can double as a somewhat common real name (ex: Jay and Rose). Even more bonus points if it's only three or four letters long. ❤️
I'd like to eventually get a name that starts with every letter of the alphabet, but only if they genuinely interest me. Right now I'm only missing U, Y and Z.
Much like I tend to get really hung up on those H and S-names though. (I also have six H-names and five S-names.)
I prefer rws above all else. 12/15 of my pets are rws and one of the 15 is a real word that's misspelled (Heavenli).
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I definitely collect names, as they're what determines if I can write for a pet. I just go for words I like and/or can think of a story for, primarily - Daylily, Parent, Seaglass, Graveyard, and Unsaid are all because I immediately thought of something for them (and the first four of them are done!), while Truly, Lissa, Lilt, Piece, and Contradiction I have because I either thought of a very rough idea that I like a lot and/or I like the word enough to make it worth it to hold onto until I do think of something. Lissa is my one exception to the RW rule. That's just my favorite name/nickname.

I have the same wish of owning at least one Pet for every letter of the alphabet one day but there are certain letters that just kill my nerves. With my new slots I finally found a name starting with X that I adore (Xandro) but how on earth I shall find a Q or an U? ;A;
Man, I feel your pain there! I haven't had any problem finding Q-names that I like, but U plagues me. I can't think of anything that interests me.
I'm not particularly fond of rn's, but I do love rw's. I think there's more potential in a name like "Stone" versus a name like "Stephanie". However, I don't have many pets at the moment so I don't know if I'd call myself a real word collector, maybe a just an admirer xp
I've been stuck on letters before too, U came into my hands by chance but X, Z, and R took some searching. A, B, and M are the easiest for me to love and I've had 6+ of each of those in the past, still several of each now.
I'm more fond of made up words and maybe real names, usually non-english names are easier for me to work with. I've got a couple real word names right now though and I'm gonna give them my all. But for the most part I don't keep track of that, as long as they look nice in my collection overall, which tends towards odd names.
all of my pets go together but in the big picture they are all part of one big plot that i cannot write. im terrible with writing and i kind of feel like my 'plan' is a little mundane, but i havent had anyone else to ever bounce ideas off of or anything so here's the family tree: much and mish are fat sisters, their toys are snookums and lamby (tech both fan pets too). much's 'boyfriend' is gast, who is best friends with 03. h8 is the bully who makes fun of much and mish and gast. v8 is h8's sister who is a polar opposite. dace and vace are also sisters with polar opposites, but dace is h8s best friend. feud is the burning desire to make peace : ) /breathes
All the tribute names!! Been on a crazy rampage for Final Fantasy monster names though. My collection is starting to get a little out of hand... ^w^;; I try to have something for all the random crap I'm a big fan of.
Final Fantasy names
Coeurl, Cactuar, Tonberry, Ochu, Sahagin, Zodiark & Ascian
Tribute names
Sony, Microsoft, Northrend (World of Warcraft), Vulgrim (Darksiders), Zamtrios (Monster Hunter), Zero III (999), Blasto (Mass Effect), Demi-Fiend (SMT: Nocturne), Derpy (MLP), Witch-King (LOTR) & Okuni (Samurai Warriors)
First off I loooove all your pet names, they're amazing. And your stories are so well coordinated. I'm impressed how verything ties in.
Ahhhhh I'm so amazed at your beautiful tribute names,and oh the jealousy of your fantastic FF names. Cactaur! Coeurl! Ascian! Love it so much.
thanks! i was really worried i was being clique but you make me feel a little bit better about it :) i just recently created v8 and vace
I'm pretty proud of myself for snagging a few names, they're mostly just RW (obscure) planty things. Beech, Protea, Neem, Fava, Balsam. then I have a few others on the side, like Cyclades (the archipelago), Entertain and Nor.
I don't have that many pets (yet!) but I definitely like RW names better than anything else. Don't really foresee myself straying from that.
For a pretty long time I stayed away from proper nouns, and also within one 'verse; Tiercel, Avow, Bravado, and Pique are all named because the words remind me of the characters they represent in one way or another. Just today I made Ciabatta and Havarti to satisfy my craving for food-based pets (Ciabatta's a baker; Havarti's a cheesemaker, as you'd expect), and I'm quite fond of how the proper-noun RW names work for them.
Also I think I'd like to have unique first-letters as long as possible--like, my current pets take care of A, B, C, H, P, and T, so for future pets I'd like to avoid those letters and pick names starting with letters I haven't used yet. Not sure how long that one'll last.
I have a mix of made up names and real word names. Since last year I've had a goal of having at least one pet name starting with every letter of the alphabet. I'm almost to that goal, I just need an X and Y name which is proving to be tricky.
I've got one RN pet and ten eleven RW pets - I definitely have a preference. :P
I've collected most main/major characters from Revolutionary Girl Utena. I started on a whim with one or two and just kept going from there haha.