Welcome, Camminatore!
Regarding the midnight start: I have been told by National Forest staff that you can hike up to Lone Pine Lake without a permit. It is only beyond the lake, and the Wilderness boundary, that you must have a valid permit. So I suppose you could start at 11 p.m.
But that would be foolish -- the sleep deprivation will catch up to you before you reach the summit. And the summit is only half way! Lots more people leave at 3 or 4 a.m. You won't be alone.
All the info about acclimatization is on the mark. Here are two recent discussions worth reading:
Anyone know about a High altitue B&B or Campground? recomendations for an acclimatization route up the summit? As posted, I was told just the opposite by the people within the same building when it comes to starting time. I have found it best to just get up and go if you can't sleep. 2 of our last 3 we were up at 12:30 and moving towards the summit at 1:30. I, like many, a couple of things working against me sleeping the night before a summit attempt...elevation and anxiety.
Those two time were from just under 12,000' but I have problems even at 8,000' from time to time. You couple that with the excitement, sleep is very difficult to come by. So, rather than toss n' turn, go! You can always catch 40 winks once the sun comes up...and we have.
I too have been told both answers for a starting time, but let's be honest here.
1. There is NEVER going to be a ranger checking permits at or near midnight. They don't have the staff.
2. You CAN NOT be ticketed for not having a valid permit below the sign.
3. If somehow there IS a ranger checking permits at or around midnight AND he/she does give you a ticket BELOW the sign, enjoy your day in court making them eat crow. You can legally hike up to the sign any time day or night. You must have a valid permit to enter the Whitney Zone (past the sign).
Start anytime you want. Once you go past the sign you are required to have a permit. Odds of getting it checked before breakfast are slim. I have slightly more "legal" summits of Whitney than "illegal" no permit summits. I'm older and wiser and try to always follow the rules now. Still, I am rarely checked for some reason.
Last year my first day hike and my MR trip were without the "Cattle tags". They didn't have them at the ranger station yet so they sent us out old school with the paper permits to put in our pocket. We passed many rangers and were never asked for a permit. Though the MR was with RichardP and I don't think they check him much.

What I use for a successful day hike is simple. I get to the Portal as early as possible the day before. I try to make it in time for the pancake. I nap and eat lunch at the Portal. I BS with Doug and the regulars as much as possible and then get another nap. If it's just me or me and the boy we roll out around 3 or 4 am. (We usually sleep in the parking lot behind the store but you need permission to do that) If I'm leading rookies I get them up at 0100 and on the trail by 0200. I lead until we pass LPL so the pace isn't too fast. Then I get the group's weak link to lead us to Outpost. Before Outpost I brief on respectful use of headlamps and we blow through. I keep the group together, always. One person fails, we all fail. I try to hike for 50-60 minutes and rest them 5. We take a good 15 minute break at Trail Camp and make sure everyone is ready. Then I lead us out and up the switchbacks to set a slow pace. I try to go 30-40 minutes and rest 5. I keep everyone snacking and drinking during those 5 minutes and drinking the whole way up. If it's not too windy we take another 10 at Trail Crest. Then 10 minutes to rest at the summit and 20-30 for pics. Then we head down.
It's never a fast trip, but so far I've been 100% successful no matter who I take.
Leave when you want, hike slow and easy and take some pics. Your body is resting while you stop to take that pic.

Remember - there are 24 hours in a day. Use as many as you need. Assuming the weather allows of course...............DUG