Why Nidorino & Nidorina evolve via Moon Stone? Clefairy, Jigglypuff and others are somewhat related to moon, night, or sleep. But I can’t find the connection with the Nido family.

Personally, I think it’s because they’re supposed to be – very loosely – based on rabbits (definitely the weirdest fµ¢£ing rabbits I’ve ever seen, though, I’ll give you that; maybe they’re influenced by jackalopes or wolpertingers or something).  Rabbits are associated with the moon in China, Japan and Korea, because the shapes on the surface of the moon that we see as “The Man in the Moon” in culturally western countries are traditionally interpreted as a rabbit, often with a mortar and pestle, in east Asia.  There are a bunch of different stories about exactly why the rabbit is up there and what the mortar and pestle are for, but the fact that it’s a rabbit is pretty well agreed, and this gets referenced a fair bit in Japanese pop culture (Dragon Ball had an anthropomorphic rabbit villain who wound up imprisoned on the moon; Sailor Moon’s real name, Tsukino Usagi, is a pun on tsuki no usagi, “moon rabbit”; Digimon World: Dusk gives the player a partner Digimon who is a rabbit with moon-related powers; etc).  It’s a little tricky to be sure that’s the intent, but it’s the best explanation I can think of.