LAX has a lot of closeby hotels. Sheraton, Hilton, Westin, 4 points which I think is a Sheraton chain. You can see these as you leave the airport they are just at the edge of it. There are some more, all in the same spot that all seem to be nice chains.
We stayed at the nearby Holiday Inn last year and it wasn't too expensive and we found it comfortable. They are on the shuttle line that runs between the terminals and the various nearby hotels.
Found them to be very helpful when we needed a few things (such as extra cribs and the like). Had a nice little restaurant in the lobby or if you want to go the fast food route there is a McDonald's next door.