We do a 20 hour drive every summer with 2 kids (now 6 and 3). The night driving would be the best unless your kids don't sleep in the car (my 6 year old has a really hard time sleeping in the car no matter how tired she is). Plus, I'm such a backseat driver that I can't sleep while dh is driving, then when it's my turn to drive I'm exhausted! LOL And, those reallllly early morning hours just before the sun comes up tends to be pretty tough on us. The one year I started doing that "I'll just close my left eye for a second, now just my right eye..." That was the last year we drove at night!
What we found works best is leave in the morning, and put in as many hours as we can the first day. We try to do at least 14. That way the second day isn't as long, because every trip we've ever taken, when we have to get back in the car the second day, the kids are NOT happy! There tends to be a lot of whining and moodiness that second day, whereas the first they're just excited to be going on vacation.
That said, we are driving down to Orlando next week from NJ. What we're doing is leaving after dinner on Monday, driving until we're just past DC and stopping. Then the next day we'll drive until we get to Orlando, find a place to stay Tuesday night, and check in to the Poly Wed. morning. It's kind of opposite what we normally do but I just want to go thru DC when it's not as busy, plus that gives us the full day of the parks on Wednesday vs. getting there mid/late afternoon that day. On the way home we'll do our normal full day the first day, that would put us going thru DC mid morning the second day.
Good luck!