An RTS might be fun. Not quite sure how it would work, but it might be fun.
Another idea I had was a Theme Hospital-type game, but based on a Pokémon Breeding Centre. The reception accepts the Pokémon, you have a building with several rooms (kitchen, sick bay, breeding room, storage room, computer/teleporter room, Pokémon training room, staff training room, etc.), and then you have a large outside area divided into different kinds of terrain where all the Pokémon are kept.
You can terraform parts of the outside area, and expand into new areas once you buy them. You gain access to more kinds of rooms, tech and staff skills as you progress through the game. Staff members include receptionist, mechanic (takes care of the tech), breeder (with specialities in nursing, tutoring, nutrition, coordinating) and scout (a trainer that occasionally catches Pokémon for the Centre).
The trainers dropping off their Pokémon can have all kinds of requests - level it up, evolve it, teach it move X, make it obey orders, make it more beautiful, etc. etc. You can also sell Centre-owned Pokémon and eggs for some extra cash or to free up space (trends can affect the prices, e.g. suddenly Clefairy is popular, so you can sell them for more).
Occasionally you may be approached by a disreputable group of people (e.g. Team Rocket) and "persuaded" to hand over some Pokémon. If you refuse, you can ward them off by setting your Pokémon off them (but risk them returning in force). Agree, and they won't bother you again (but you risk taking a popularity hit, not to mention a trainer will be very annoyed if you handed over their Pokémon).
There's huge scope for this idea, and if I knew how to do the graphical side of it, I'd probably have a go at it myself (I'm fairly sure I can do some acceptable coding for it). If someone wants to team up with me on this project, let me know.