I haven't tried a variety of options, really, so I can only share my (limited) experience.
Since the beginning, I updated one minute after midnight (EST) on the scheduled update days. Always the same time.
I likely sound like a broken record by now, but I made consistency a sticking point. I set a specific time, and I meet that time, readers become accustomed to it.
This does a few things for me:
* I have a hard and fast time to meet. I'm a procrastinator, and if I said "sometime in the morning" I know I'd slip to "sometime in the afternoon" or "sometime later today" and I wouldn't get it done. Having a set time forces me to knuckle down.
* As a related point, it's healthier for me, to have midnight rather than some time in the morning, or I'd be pulling all nighters to get stuff done for the morning.
* It's also easiest, because I know I'm a night owl, and this way, the push to completion coincides with the twilight hours post-mealtime, when I write most effectively. I can then stay up until 1:30 or 2 am to see the initial comments (and make necessary changes), then sleep and wake up to that many more.
* Readers can experience that anticipation as the time approaches. I value this anticpation as a reader, and I wanted to cultivate it as an author.
* Readers who wake up are guaranteed to be able to check out the story as they start their day. No waking up at 7am for a story that updates at 8, checking in at 7:45 and then forgetting about it for the rest of the day. (Not accounting for time zones, but applies to the Western hemisphere).