I suppose a good place to start would be if the U.S. were willing to return to the Commonwealth.
I mean, it wouldn't mean anything, it would just be a figurehead movement so the U.S. would have Queen Elizabeth as a useless figurehead (at first), but it would at least get the ball rolling.
I have no idea how to pitch that, however. It would have to be sold as some sort of "Anglosphere PR/diplomacy" thing, and repeatedly mentioned that, "Of course, this doesn't really mean anything, Queen Elizabeth doesn't have any actual power, etc.." But it would be really difficult to do even that. However, assuming it could be done:
-It would start people thinking about the role of monarchs again. And that discussion that seemed to close itself around the ochlocracies that sprang up after the World War I that continue to exist today would finally begin to be questioned again.
-If the U.S. did it, it would be huge.
From there, I suppose some sort of worldwide movement to at least have a figurehead monarch for every country possible would need to happen. Then it's pretty much a waiting game. I mean, all one needs to do is have something like the February Coup or what happened in Belgium 2010-2011 in order for their existence to be vindicated. Keep them small, at first, as figureheads, and then just as these "constitutional referrees."
Then…when there's a real major political crisis, they will be there to fill the vacuum.