If done correctly, that is. The whole hook of the film was putting you inside the action. If a giant monster really did attack New York (which is like, totally possible- my brother told me), it would probably happen like the movie. You know what else is good at immersing the audience into the action? Video games. By that logic, I say Cloverfield could potentially work better as a game than a film.
I'd imagine it as something similar to Left 4 Dead. It wouldn't have to follow the plot of the movie, but for the sake of this writeup, let's say it did. The party at the beginning serves as the tutorial on the controls and such. Then you get into the action with the attack, and you have to push your way through the chaos and confusion as buildings collapse and people run for their lives. The police and military are all over the place -- avoid them as they try to wrangle you in with the other refugees.
Then there's the obvious part: the dog-sized parasite monster-things. Fight them off with a metal pipe or other objects from the environment. There wouldn't be any guns or standard weapons here. The scene at the main dude's girlfriend's apartment (I don't remember her name :P) would have light platforming and you'd have to struggle to maintain your balance, since the building was toppled and leaning into the building next to it.
For the sake of atmosphere, you would be able to hear the giant monster in the distance as it rampages through the city; even catch a few glimpses of it at certain moments, and of course see some of New York's skyscrapers crumble to the ground. Police, military, and refugees would be all over the place, the sound of gunfire would be ever-present, the ground would rumble every now and then. By the end of the game though, there would be more empty areas- like that scene with the empty horse carriage- and military personnel would greatly outnumber refugees.
So there you have it; I think that would make for an intense and atmospheric game. As for the gameplay itself, ideas are loose, but I picture it being a third-person action-adventure/survival horror. Like I said, there could be bits of platforming and combat, possibly even stealth, but I haven't given it much thought yet. :P