Earlier quoted context omitted.
It's not clear that the rotation of the moon has much significance in modern life any more, so maybe the problem can be simplified. What was the significance of cycles of the moon in the past, anyway? The main thing I can think of is astrology. Can tides be predicted accurately using phases of the moon?
Month = "Moonth" Aside from the female menstrual cycle, the gestational cycle is also in line with the moon: A baby is brought to term in 10 moons or 280 days (9 months or so). You would also have an easy way for disparate peoples to mark time, since in the past there was no such thing as telegraphs or even pony express.
A human pregnancy is around 266 days +- 16. The reason for 280 days or 9 months is because that measures the time span since the last menstruation, not since conception.
(I was personally a bit surprised by this when expecting our first child. It's one of those things you apparently know about but still use "9 months" as a bit of a fixed expression.)