I wouldn't call it a frequently used idiom, but I do hear it used occasionally. I estimate I go months at a time without hearing it, however. (This is in the northeastern U.S.)
"When did you buy your tux for the wedding?"
"I waited until the night of."
There's an absent, implied "the wedding" or even "the wedding itself" -- the formulation tends to further emphasize a bit that something happened (maybe even contrary to expectation) on the very night or day of a significant event. If spoken, it helps make the idiomatic usage clearer if a very slight pause follows, to emphasize that something has been bitten off, and likewise in written text, it's generally situated before a comma or period.
I think I usually hear it describing something happening on the same say or night of an event, but in advance of the event itself.
"On the day of, we took a good last look at the house, knowing we would never see it again."
I think I've usually just heard it used with a time span like night, day, evening, or morning, but not time units like month, year, etc. Like I said, it emphasizes that something happened nearly contemporaneously, so longer time units don't fit. It also wouldn't be used with hour, minute, second, or moment.
As it's something that also appears as a substring of more common constructions, searching for examples can be probably be best done by appending words that wouldn't otherwise follow. For example, search for "on the day of we went"
However, perhaps the most frequent use is not alone but together with "the day before" and/or "the day after".