I think the reason some people might believe this claim is because we’re taught in school that the moon’s gravity causes the tides. I think the reasoning goes, “well if the moon’s gravity can affect the tides, surely it can affect smaller things too”
Iirc, this is actually true. There has been some studies that suggest that the moon phase does influence menstrual cycles. I don’t fully remember what the researchers concluded, but I remember vaguely that it was something like, not seeing the moon (if the sky is overcast, for instance) doesn’t affect the menstrual cycle, but looking at the moon will “reset” the cycle in some way.
Well, now you made me find this paper: https://www.science.org/doi/10.1126/sciadv.abe1358?intcmp=trendmd-adv
OP’s friend might have read something based on it too. This is really weird. I guess more reading is necessary to know if this paper was already debunked or not