This is what we said to each other when going to bed 60 years ago in my corner of Tennessee —not “good night”, and not even “nighty-night”.  Not just for children, but adult to adult as well. I haven’t heard it since I left Tennessee, but just the other day a Texan bid me “night-night”, so I asume it is southern (U.S.). Now that I’m thinking about it, “night-night” strikes me as peculiar as greeting someone with “morning-morning” or “afternoon-afternoon”. I can’t guess how it started, nor “nighty-night” either, which I assume is related. Anyone?

I grew up with it in New Jersey. And the OED has it from 1866 with several British citations, so it’s not a regionalism. (The big dic has nighty-night from 1888.)

It strikes me as a straightforward example of reduplication and dropping an element, like bye-bye from good-bye.

Just to confirm, also in widespread use in the UK.

And just to add to what Dave says about the reduplication and what westover says on why only for this time-of-day greeting, it’s the only one with one syllable, which always helps reduplication (as Dave’s example also shows).

But never day-day for good day.
Perhaps becuase good day is a little more formal.

But, cf. wakey-wakey.

- or “Ta-ta!" or ”‘Ello, ‘ello, ‘ello”.

I was going to compare the North German greeting ‘moin moin’, but it’s not as close a parallel as I imagined; I’d thought the word was a corruption of Morgen, meaning ‘morning’, but the majority opinion seems to be that it’s from the word mooi, meaning ‘beautiful’.

Actually, my father-in-law DID used to say “day-day”, when people said “night-night” to him. I was going to mention “pyjama, pyjama” but I see the OED records this stock response to “nightie-nightie”, quoting the Opies, famed childhood folklorists, as its source, so instead I’ll just repeat one of my daughter’s favourite poems from when she was six:

“Night-night, Knight”, said one Knight. to the other knight the other night.

The reduplication in night-night and bye-bye is clearly originally babytalk, but more interesting, I think, is what might be called “blended” farewell words, such as “cheery-bye” and “tattie-bye” (both in the OED), combining goodbye with cheerio and ta-ta respectively. This seems more “language used with familiars” than “language used with small children” - more informal even than “cheerio”, while still not descending completely from the adult register ...

Interesting, Kurwamac. I always thought that ‘mooi’ was one of those typical indigenous Dutch words of unknown origin and limited to the Low Lands only.
EWN and WNT seem to confirm this. Both say it is not found in any other Germanic language.
It can of course have been borrowed into a German dialect. Dialects don’t care much about borders.

Moin moin isn’t known to me BTW.

This does remind me of the Hamburgerian(?) exclamation “mors mors” (kiss my ass) in response to “hummel hummel”. Some background information can be found here.

Edit: maybe the fact that the Frisian word for ‘morning’ is ‘moan’ has something to do with it, but maybe it hasn’t.

A good discussion on the etymology and meanings of moin in various Germanic languages (link) and dialects.