If you stay at an LAX hotel, you might be able to avoid having to pick them up the next morning, and instead have them take the hotel's shuttle to the hotel and meet you there.

You could do the same when you arrive, save a night's cost for parking and rental car, and pick up the rental car in the morning. There are some agencies that have locations in some of the airport hotels in addition to the main airport locations; these can often be cheaper since you don't pay all the extra airport fees. You can usually arrange to return the car to the airport rather than the hotel location.