I enjoyed the story while I was hearing it, but I found that there were more and more little "annoyances" as I thought about it. The first, most obvious one, was genetic drift/evolution. How long had the May's been around? For every one human generation(roughly 30 years), they go through 1560 (about 46,800 human years)- even a small drift would show up quickly. What about other May's who left it to the last minute and ended up sleeping with some goofball loser - what "bad" genes did they get? What about mixed race babies? Wouldn't it be slightly noticeable if next week your neighbour suddenly became mullato?
Second, why would they spread out so far - granted it cuts down on the possible discovery, but if they had overlapping weeks - born on a Monday, born on a Wednesday - like the old nursery rhyme - they could help each other out easily (and it could be platonic - sorry, Thaurismunths). The idea of communal families isn't new.
Third, how long did it take her to get autobill paying? Man, I age in a much slower fashion and I got that ages ago! I'd think they would have developed a multitaking ability generations ago - why wasn't she at least reading the book while on hold?
It really works as a "remember to cherish every day" kinda thing, but the logic falls apart too easily for me to really enjoy it.
I agree with the other comments that the twins ending seemed tacked on, and almost "extra last flip" that didn't tie into the rest of the dance routine. There was no need for it.
Also, the generational memories got me to thinking that this is less like a Mother/Daughter relationship and more like an accelerated Dr. Who kind of deal. It's really the same person changing bodies every week - though instead of moulting, it more of an internal transferance. I get that the story needed to explain how an 11 year knew how to pay bills and cook, etc., but this fundamentally changed the relationship - you're not teaching me, I'm remembering being you. In effect, the Mayfly is an immortal of sorts.
To maintain a Parent/Child idea perhaps the solution would have been a high speed language of sorts, a way they could only communicate amongst themselves, and it would sound like buzzing to any human listening to it (kinda like the Star Trek episode where Kirk gets "accelerated").
Quick aside - in three postings to this forum today, I've referenced three different ST:TOS episodes - man that was a great series!