You start off with the obvious ones:
And gradually start adding the more obscure ones (and a few obvious ones you forgot):
Depending on your personal preference, you may want to list subgenres (i.e. FPS breaks down into: Twitch Shooter, Modern FPS, Man vs horde games (Serious Sam), Survival Horror (RE4), etc.)
When you hit a wall, start thinking of existing games and try and categorise them:
- Dungeon Keeper 2, Black and White - God Games, RTS
- Bit Trip Runner - Rhythm
- Civilization - Strategy Games
- Portal - Puzzler
- DotA - MOBA
- Capsized - Physics Game, SHMUP
- Limbo - Psychological, Platformer
- Diplomacy - Negotiations
- Shadows over Japan - Information Management
- Liar's Dice - Bluffing, Wagering
- Dominion - Deck Building
- Pokemon - Fighter, Adventure, Collecting
- Diablo - RPG, Dungeon Crawler
As your mind wraps around them, you will gradually start thinking of hybrid categories and various combinations that would be interesting.
You may also start combining genres in your mind and coming up with examples of such things:
- RPG + Rhythm + Adventure - Brütal legend
- Information Management + Puzzler - Salem
These are generally distracting and less than productive, unless they can give you new ideas. The thing about looking for inspiration is that you never know where it is going to come from. This is an exercise that I have found helpful on many occasions.
Depending if you have come up with anything yet, you may want to start just combining genres at random and figuring out how (and indeed IF) they would work together.
Would a Collecting/Puzzler game work? Collect various pieces throughout the game, and use them in successive puzzles. Depending on the mechanics of the puzzle, it might end up playing something like Shogi (where you can either place a piece or move one; though the game itself is most similar to chess) where players alternate positioning pieces to outmanoeuvre the other and pick at the loose threads of their pieces.
How about a fighting/racing game? Race against your opponents and damage them to slow them down. It would require ranged attacks of some kind, so why not go all out and mount machine guns on the cars, and turn it into a full-on demolition derby? Because cars would take huge amounts of damage, include some auto-repairing features, and a shield to mitigate damage at the cost of speed (though less than what you would lose for taking the hit outright).
Still unsure if an idea is viable? Test it out. Build a simple prototype, and playtest it. Success is 10% inspiration and 90% perspiration. Even if an idea is good, it will take many, many stages of refinement to turn into something that you can be proud of. But in the end, it is worth it. But more on that next time.