I'd recommend staying in Santa Monica.
Not too far from LAX (as you will be heading up w/o a car).
The Avis office is open on Sunday.
It's a good drive from there as you will literally be on the 1
And, you have plenty of restaurants/nightlife
Here's the Avis link:
http://www.avis.com/car-rental/locat...CATIONCODE=SM8