If you have a very early AM flight, try staying at a hotel in your homecity near the airport.

This allows you to do a few things: earn more nights/"mattress runs" with your hotel point program, park cheaper than airport lots, earn points for your parking, and of course be already near the airport and avoiding traffic delays for that early morning flight.

Example: FourPoints LAX has a park/stay rate that includes 1 night stay, plus up to 7 days of parking for $119.

parksleepfly.com has listings in many cities.