We drove straight through when I was 16... almost 20 years ago. Eek! We drove from Scranton, PA.
Anyway, it really wasn't a problem, well, except when my Mom kept telling my Dad that she thought she should find a gas station and he kept telling her that she had plenty of gas left even though the gauge was on E. We were stuck on the side of the road from about 3 a.m. until 6 a.m, when a very nice truck driver gave us a gallon of gas to get to the next exit.
AAA wasn't an option then since no one had cell phones back then.
Anyway, we just kept alternating between my parents and me with the driving and we all slept well when got there.
My DH and I drove through the night once to Chicago, too. We had worked all day and had a business dinner that night. We decided that instead of going home to sleep for a few hours and start at 2 a.m., we'd start at 10:30 p.m. We alternated driving every hour so we could sleep, too.
The greatest thing about driving through the night is NO TRAFFIC! I think that psychologically, you know that the only other thing you'd be doing then is sleeping, so it helps the time go by better. We drove back from Chicago during the day and that was torturous because of all of the "good" things we could have been doing instead of driving. (We flew the next time we went to visit his friend there.)
IMHO, everyone should experience at least one long distance driving trip in their life. I've taken many and they sure do give you an appreciation for air travel. lol