DH flies through LAX several times per month, occasionally on Saturday. In his experience, the airport can be very busy on Saturday mornings. We've done LAX before from Anaheim and hit traffic on a Saturday morning. It took us about an hour and 15 minutes to get to the airport.
A conservative plan would give extra time to account for freeway traffic, car rental shuttle waits and long lines at the airport. We allow no less than 1.5 hour between ticket counter and arrival at the gate, 30 minutes between car rental drop off and getting to the ticket counter and 1.5 hours for traffic. This gives us extra time to catch a bite at the airport before flight, too.
If you go by this plan:
Departure time: 12:40pm
Time needed to be at gate: 12:10pm (plane loads at this time)
Time needed to be at ticket counter: 10:30am
Time needed to be at car rental dropoff: 10:00am
Time needed to depart hotel: 8:30am