I can't imagine the US ever becoming a monarchy. But if that were to come to pass... I'd say that we steal someone from the British Royal Family, simply because they are the most well-known Royal Family in the world and, once upon a time, before the Revolutionary War, the original 13 colonies were British. But, it couldn't be just any random, lesser member of the BRF, like the Queen's cousins. Maybe Edward or possibly Anne. Probably not Andrew (too many scandals and Fergie) and I can't imagine Harry would accept the job.
If not the BRF then the next option would probably be the Grimaldis of Monaco. The theory, I'm guessing, would be that they at least have some American blood in their veins thanks to Princess Grace.
And then, I suppose a third argument would be someone from the Hawaiian Royal Family, if anyone is still alive.
It's an interesting question, though entirely, completely hypothetical.