American politics has not needed a third party badly enough to actually form it. If and when the two-party system fails utterly (and it might, conceivably, do so in the near future) a third party will arise on its own. Given the present circumstances, a third party would probably be a moderate party that favors a liberal social agenda, fiscal conservatism, doesn't fear government but doesn't love it or trust it particularly, and tends to realism in foreign policy.

Third parties have failed in America because one of the two major parties has usually corrected course to seize the moderate middle in any election, and it has worked just fine for over a hundred years. If, as some are saying, the parties are too polarized to move back to the middle, then a third party will arise, but it is likely that it will just absorb the party that lost out the most from its formation.

However, it wouldn't be that hard to create a new party. There are enough rich people in the US who would favor the platform set out above that you could quickly find moderate candidates (and even sitting legislators) to break ranks with their former party and join the third party movement.