Yeah, I've run out of good name options because I have so many pets! It's sad that we can't just name them what we want to. Here's a list of names that could be added. I don't expect them ALL to be added, but some of them would be nice:
Acorn Arrow Ami Berry Bonnie Brownie Bonbon Butterscotch Cackle Cake Candy Carrot Caster Chaos Cheddar Cherry Chirp Chompy Coils Crystal Dash Daybreak Dew Destroyer Dizzy Domino Dutchess Dusk (I'd really love this one.) Ellie Eon Fuzzle Gin (Means silver in Japanese) Grizzle Howler Iggy Jumper Kane Kaos Lancer Larry Laser Lasher Lizzy Magic Moonlight Nomnom Nugget Obsidian Otto Pal Paws Peep Pickle Pocco Puffy Puppy Quack Quartz Queenie Quills Quiver Rainbow Raisin Rustle Sabre Salty Sparkster Tabby Taco Tangle Uno Valkyrie Vanilla Velma Violet Waffle (I think a lot of people would like this one.) Wally Whisper Wish Xeno (Xena is available already, I know.) Yang Ying Yumi Zilla Zuzy
(Long list, I know, so if anything's already available in-game, just ignore it.)
Alfie Alpha Ash Bell Carol Dream Dusk Echo Excalibur Flash Fluffy Gizmo Gypsy Hope Hugo Ink Iron Ivy Jade Keira Key Kirk Lance Lion Luna Marshmellow Mist Nelson Nimbus Nina Oak Onyx Oscar Oz Pacific Pal Patches Pirate Pit Quartz Rex Rocket Rosa Rose Ruby Saphire Spade Sparkle Storm Sunny Sushi Taboo Tango Toxic Toxin Travis Treasure Utopia Vanilla Voodoo Waffle Wanda Walnut Watson Woody Xina Yam Yang Yin Zero Zeus Zip Zoi