The one I first heard and adopted personally looks like a combination of others mentioned:
“Dark o’clock.”
That, and a phrase I’ve pieced together from other sources: “If God wanted us to enjoy the early morning, he would have scheduled it for later in the day.”
Okay, I confess - I really am an early riser. I promise I’ll be quiet while you’re still asleep.
I associate oh-dark-hundred with MAS*H. I believe Henry Blake said it once or twice.
As a Ren faire participant, I’ve heard “gate o’clock” (when the patrons leave) and
“naked-thirty” (when you can remove your garb.) In real life, I am a night owl rather than an early bird, so I refer to those times I have to get up early as “oh shit o’clock” and those times I’m up too late as “It’s ten past stupid.”
Having pets who very much abide by routine, we in my household have noted that an alarm of sorts goes off at cat food o’clock, and it is wise to be punctual in releasing the hounds at poop o’clock.
“Not sure what time zone you’re in, but it’s beer thirty here.” I use that one either when I’m travelling on business or when I’ve got business visitors in town. Eg, about once a week
I say oh-dark-hundred to refer to any stupidly early hour, as in “I have to get up at oh-dark-hundred to get to the airport.”
Another useful time is one that falls during my friend Drew’s parties: pantsless o’clock, the hour after which most people have their pants off.
Dark-thirty here.
But I’m using it to indicate after dark evenings.
I’ve heard it, but I prefer “The crack of ‘Why am I awake?’” to describe early morning before I’m ready to get out of bed. 5am counts as this and “Oh-dark-thirty.”
I used to work with military types. That’s where I picked it up. I love it.
Another variant I’ve heard is oh-god-thirty, also to reference an entirely too early time.
I didn’t pick up that militaryism from coworkers, although I wholeheartedly embraced “charley foxtrot”.
I describe that time of day as “the buttcrack of dawn.”
O-dark-thirty here. Never really thought about it - just a more fun way of saying “I have to wake up earlier than any sane person would.” Dad is a vet.
I think I like your friend Drew.
(But they’re all gay, aren’t they? Aren’t they? All those beautiful pantless men and I couldn’t enjoy a single one of 'em. le sigh)
Another sailor voting for oh-dark-thirty.
Of course, when we had the midnight-to-4am watch that was referred to as the “balls-to-four” watch.
Former Squid here. Used Oh Dark 30.
The Buttcrack of Dawn was also popular.
right up there with ‘big ugly fat fellow’ huh? 
another dark thirty user, here. had to have picked that up from all of my police officer friends. never heard of ‘beer thirty’ before, but of course i shall immediately appropriate it and use it as much as possible from here on out. 
Never heard it, but I haven’t spent time around military people.
Another Air Force vet* huh?
Oh dark hundred was popular in our family for a long time.
For those not in the know, the B-52 Strato-fortress bomber is also know as a “buff”. While some folks may think it’s a tribute to the camo color scheme, it’s actually based on the fact that a B-52 is one Big Ugly Fat Fucker!
I regularly use
FWIW, I’ve never been in uniform, but I’ve been working beside them most of my career life.
Slight hijack, in Hebrew something like this is built into the language – times from 2 or 3 AM to <whatever is too fing early!!* for you> are, officially, X “pre-morning.”
This is especially useful when the night-owl at work is late for the 10 AM meeting, and when his absence is noted someone will always pipe up with “waddyawant? You set a meeting for 10 pre-morning and you expect him to be here?” 