Okay, it's only the second idea and I'm already resorting to thinking up platformer games. I'm not really out of ideas, I just think that platformers are such a great game type because most people have played one before, and it's so simple to learn how to play even if they have never played before. Essentially then, platformer games are a blank canvas on which you can put new, unique features and mechanics.
So my idea: A basic puzzle platformer, where you have to collect coins and keys unlock doors etcetera. But the twist is that wherever you walk you leave a faint ghost-like trail behind you. The trail does nothing except look cool - until you die. When you die you have restart the level, but the trail turns solid. Every time you die the last trail you created disappears, in order to prevent the level becoming a tangled mess. There could be a 'self-destruct' button so that you can arbitrarily decide when to create the solid trail, and completely restart the level if you need to (by self-destructing at the very beginning there would be no solid trail next time, because you hadn't moved to create a ghost trail yet)
So, this trail system is great an all, but how exactly does it make the game more fun, difficult or interesting?
Well think about this situation: there is a bottomless pit of brimstone and hellfire and you just can't quite jump all the way across. By jumping in once, your trail creates a bridge that goes half way across the pit. You are then able to walk on this bridge and jump off it to the other side of the bridge.
My friend No-Eyed Pete does a better job of explaining below:
Another challenge could involve a platform high in the air that you need to get to, but you just can't jump high enough. By jumping up and then self-destructing, you have created a nice step on which you can jump to get the rest of the way up to the platform.