I'm way too lazy to read all the replies, so my bad if this has been said already, but INCEPTION FTW! Seriously, it would be an amazing game. It should have its own storyline, but keep the world and the tone of the movie. It would probably be a regular third person shooter in terms of basic game-play, but going on some of the mentions they made in the film, it would be awesome if you could have conversations with the projections and try to discern the info you need from them. Or you could convince the guy who is dreaming that he needs to tell you. Or you could go the sneaky route and use the safe idea, where you create a secure location and break in. If you are discovered or break laws or simply act too out of place, it could raise a suspicion level. If it gets too high, some of the projections attack you, like cops and the like. But as it climbs even higher, they will start to be able to sense where you are (this would work practically by telling the AI that you are somewhere in an area, which gets smaller as the suspicion level rises). If it gets really high, even regular civilians will attack. If you fill it up, every projection will attack and will always know exactly where you are.
The way you can get out of hot water is to either create weapons or equipment using your mind, or to activate special fail safes installed by the architect. These would be nutso things, like making walls appear or disappear, or raising bridges or creating lava flows to kill enemies. They would be super powerful contextual abilities activated with a button press, but they raise your suspicion bar a lot. There could also be a level maker mode, IE, a mode where you are an architect. The reason the game would be good is that it fits in lots of videogame staples and necessities into the world of the game so they seem natural. For example, what's actually going on in multi-player? Oh, those are the military training sessions that the movie called the first actual use of the dream world. It all fits. Plus, having a good story wouldn't be hard given the mind-bending sci-fi ideas you are already using. I was thinking it could be that you are a detective, and you break into criminals minds to find out if they really did it, so you know who to look for evidence on. Or maybe you could go the less morally upstanding route and be a criminal who uses inception to make other people commit crimes for them.