I have a few ideas, most of them based on other games though.
1. You're Ronald McDonald at a kid's birthday party, when suddenly the Burger King bursts in through the living room wall with all other fast food mascots behind him, and he tells you he's gonna kick your ass for making kids eat horse meat - if you don't defeat all other mascots and the King himself; but if you do, you'll become the new "burger king". You also unlock the other characters, new outfits and stages, and a new title screen. But you start out with Ronald in his typical clown outfit and fight each mascot in his own stage; or you can do Adventure Mode, wherein you race across different fast food themed levels before fighting the level's boss (a mascot); so Adventure Mode is just like the other normal mode but with levels. Beating the game with all the mascots unlocks secondary fast food characters like the Hamburgular, who can help out in fights (either automatically, when you press a button, or when your character dies) and unlocks the level editor, so you can create your own stages and levels, than use them in the main modes (in addition to or in substitution for the regular stages and levels) if you want.
2. You're Lucy Joyce, a street racer who's just washing her car, when suddenly you hear about Green Light Island: a far-off kingdom where legendary racers live and compete for treasure, money, and supremacy. Divided into eight worlds: jungle, circus, factory, mines, volcano, large lake, snowy mountain, and desert. Six race courses per world, with a reigning champion at the end of each world. Race courses are platformed, so you're not just racing but going off ramps, dodging enemies and attacks, jumping from platform to platform, bouncing off other racers, etc. During races, you can pick up add-ons for your character and car (costumes, boost power, etc.) along the way and use items (oil slick, missiles, etc.) to help you win the race. Between races, you can customize your appearance and your car's stats via a dress-up game interface. The final boss is a multi-track-drifting train conductor, whom you race across the island while being chased by dinosaurs and lava.
3. You're Red the Fish, a mutated fish with humanoid intelligence and abilities who's out for revenge against the scientists who captured and performed terrible experiments on you that resulted in your mutation. A beat-em-up where you fight scientists, robots, mutated sea creatures, and a supercomputerized security system through an underwater laboratory using an array of weapons (laser guns, mini-submarines, exploding fruit, etc.) that you can customize and that, depending on the number of kills, can evolve into better weapons (ex: laser gun evolving into laser cannon), or you can stop them from evolving by pressing a button. As you go along, you befriend and free other fish, who can take Red's place; and during multiplayer, you can have a whole school of fish fighting with you. When you destroy the underwater lab, you fight the final boss: a massive military helicopter loaded with missiles and laser guns. After you beat the game, you can play again on a harder difficulty.
4. You're a young boy who's ready to become a mecha pilot on Data Isle, so you start off on your adventure across the land: finding new parts for your mecha, battling other mecha pilots for money and more parts, gambling for more money, and buying and selling and bartering items. Your dream is to become the greatest mecha pilot, but as you go along, you find out Data Isle is ruled by a ruthless mob boss who's also the top mecha pilot in the world. You beat his goonies along your journey, learning about the hidden beauties and the dark secrets of Data Isle, until you come to the mob boss himself. You beat him in a straight fight and become King of Data Isle, and are informed of a top secret mecha the boss was working on underground. If you choose to go check it out, you'll find a mecha piloting a mecha, whom you'll have to fight. After that, the game is over: you can keep exploring, battling, buying items, etc. if you want. You can customize your mecha inside and out: changing stats and the mecha's parts and colors, as well as drawing the cockpit using a stylus: so if you want to draw a window, a button, a chair, or whatever, it'll immediately become a new 3D object in the cockpit, and you can decide if the object is to function as a window, a button, a chair, or whatever.