My ideal pet breeding game would be adult-oriented; it would not allow anyone younger than eighteen to sign up. I've thought about things like this in the past, but lack the money, talent and experience to do anything of the sort.
Genre: Breeding Sim + Interactive Fiction/Visual Novel Hybrid
Theme: Eldritch Horror (it would be pretty cool, I think, to have a breeding sim where you breed Lovecraftian-style monsters, instead of your standard, adorable pets. Of course with certain marking and mutation combinations, it would be possible to breed horrifyingly precious little Cthulhuwus!)
Breeding System: At its core, it would be rather similar to Wajas: the offspring's litter size would be randomized; the offspring's colors, markings, marking strengths and mutations would depend on the parents' genetics and there would be the possibility for "carried" or "recessive" markings and mutations to occur. Spontaneous mutations could potentially happen on rare occasions, which would make higher-generation offspring valuable (kind of like how multi-gaze and primal eyes can happen on Flight Rising.) I prefer complex, interesting systems to anything else.
Pet Customization: Players would be able to buy custom tokens with varying amounts of markings/mutations (as an example: a token with eight genetic slots would cost signifcantly more than one with two slots) to create Gen. 1 (or Custom) pets; it would be a premium item that would cost USD and would allow them to pick the number of markings and/or mutations that the token permitted. Each breed would have its own creation tokens. A rather large number of genetic items would be purchasable from a standard shop with rotating, seasonal stock (it would use standard currency.)
Premium and Free Accounts: Yep, there would be free users and the ability to upgrade accounts. Upgraded accounts would have their own perks and players would be able to purchase upgrades that last anywhere between a month and forever.
Events: It would have monthly events, sort of like Lioden or Wolvden, where there would be a certain "theme" for each event that ties into the site's overall storyline. The mechanics for each month's event would vary, that way it didn't get old or boring: some months would have a breeding contest tied to the event, others could involve fighting, hunting, gathering, cooking, crafting, solving a cipher or mystery, et cetera. Not all of the tie-ins would be a competition, of course; many of them could be thought of as a quest of one kind or another.
Automated Holidays: Like Wajas and maybe a few other games, there would be some automated holidays, like how Wajas handles Santa Apollo and Trick-or-Treating. The players' Christmas trees and birthday cakes are neat as well. Many players like to celebrate their on-site anniversaries as well, so adding something like Lioden's anniversary gifts would be pretty neat.
Interactive Fiction Integration: The site would need to have a lot of worldbuilding and lore, with story additions on roughly a monthly basis; rather than just being a static lore piece, the players would be faced with their own choices that would affect different things. such as their "alignment", "reputation," what deity (if any) their lair follows, et cetera. Choices they make during the monthly story updates would offer different monthly awards, grant different account titles, possibly even unlock special exclusive items, shops, breeds, quests and the like. Of course, the story elements (other than the tutorial) would be optional, for anyone who wanted to remain neutral, or wasn't interested in becoming part of a specific faction. (Think: Flight Rising flights, without being limited to a specific eye variety + with actual implications.)
Social: The standard stuff: on-site inboxes, forums, the ability to leave comments on other players' profiles.
Mini-Games: Mainly standard stuff. Daily login streaks (currency-gaining), slot machines, trivia, lotteries, raffles, blackjack, memory, match three, beauty contests, NPC-fighting (like FR's Coliseum, or Lioden's exploring,) cooking, word games (Hangman, word searches, crossword puzzles, etc), bubble poppers, block-breaking games, anything like them would work~
Other Features: Vacation/Away Mode (would flag a user's account as inactive, basically "freezing" it until they came back), player-run auctions, raffles and trades.