Try M Cafe de Chaya at Melrose and La Brea -- it's not that far from Beverly Hills and you have a good chance of a star sighting. Another good choice is the Newsroom Cafe on Robertson across from the Ivy -- healthy and near all the trendy Robertson shops.
There is also the Urth on South Beverly which is much better then the one on Melrose. (less of a scene and none of the out of work commercial actors) I actually think the food and service it much better at this location and the seating it better as well.
Jack Sprats on Overland & Pico -- they serve warm pretzils in place of bread and have terrific air baked french fries. When I lived on the west side this was my staple -- great salads, turkey burgers, etc.