The island continent is more immediately accessible and usable, but I think having a tiny second moon show up would give a tremendous boost to space exploration and development. IMO that's better in the long run.

I wonder what an additional moon would do to the tides? Not to mention the female monthly cycle? Hmmm. Given the chances of twice as much menstruation and early onset menopause I am going with the new island.

It would have no impact on the menstrual cycle. There actually isn't a statistical correlation between the current Moon's lunar cycle and menstruation, so throwing a new one in isn't going to change anything.

I expect there'd be some tidal effects, but probably not very strong ones unless the asteroid was in an unreasonably close orbit (which probably wouldn't be stable, so that'd be ruled out by the terms of the WYR).