If you are at all into beer, there's a pretty decent beer bar in the Sheraton near LAX (it's on Century, right down the street, like a 2 minute cab ride, or the Sheraton probably has a free shuttle): www.fourpointslax.com/brewsters-bar-near-lax
Dockweiler Beach is kind of on the backside of LAX and pretty nice. Manhattan Beach is a cute village area where you can get a late lunch and walk around looking at shops and also walk along the water. It's probably 15 minutes south of LAX.
Randy's Donuts and the In-n-Out near LAX are the staples of people hanging out near the airport, but you have 12 hours. My SIL had an 8 hour layover at LAX once and we picked her up and were able to do a lot more than just hang out by the airport (we went to Santa Monica and later our apartment in Hollywood). Actually, Santa Monica would be cool, too. You can walk along the beach/boardwalk, walk the shops at the Third Street Promenade, and there's a big mall at the end of the promenade, too. I also suggest Border Grill (at Main and 4th, I think) for a late lunch, in that case.
@kirkette probably has some suggestions for that specific area, too.