The Queen has immense theoretical powers, but it would trigger a major constitutional crisis if she were to exercise any of them without being asked by the PM.
At this point I'm not sure if UK can do much to avoid a major constitutional crisis one way or another. The deal would be seen as betrayal of Brexit by most its supporters, and remaining even more so. Hard exit would make a mess in Northern Ireland, for starters, and then you have Scotland, and not to forget that Remainers actually make the majority of the population in UK as a whole at this point.
The fun one is the right to shut down the entire house of commons, including the prime minister. That leaves her running the whole show pretty much, though oddly without the right to raise funds via taxes.
