Madrid airport

Alcala? An attractive town and home to Cervantes. It's not far off the direct route from Valencia to Madrid airport and only 30 minutes drive to the airport. But if you need to be at the airport for a flight leaving at 6 AM and have to drop your car at some point, don't you really need to get there the night before your departure, so Alcala would be a short visit during the drive, not an overnight stop? In which case perhaps somewhere en route that splits the journey more evenly (Alcala would be towards the end). Perhaps, Tarancon? It's worth a few hours since you're passing and has a nice gatehouse. Belmonte would be more interesting, but not quite on the direct route.

Edit - I assumed you were asking for places whilst driving between Valencia and Madrid airport, not places within Madrid. Perhaps you might clarify?

Barajas Airport is a distance outside of the city centre of Madrid, with such an early flight I would not hazard staying in the centre Madrid, best stay at a hotel near the airport. Also I'd say it's best to drop off your car the day before, so you will not get bogged down by driving though Madrid/dropping off the car the morning of your flight.

I believe the OP wants to spend the night somewhere on the way to and not far from MAD (and not in Madrid itself), then drive to the airport, around 2.30-3.30 am for a 6 am flight.

Another vote for the very pretty historic town of Alcala de Henares. However, I have never dealt with a rental car in Europe and don't know how the need to return the car and get to an airport hotel should affect your decision about a place to visit on the way.

If returning the car the day of departure, ensure the rental agency office is open that early. Personally, I don’t feel the signage to get to the rental car return was well displayed and on more than one occasion have passed the return lot and had to go around again to get to the return entrance. The new return lot, near the old one, is a little better marked, but the signage to get to it isn’t much better and would be harder to find in the dark. It’s not that there are no signs, they were just small. When I asked the gentleman at Avis, he stated that’s as large as they were allowed.