Just two blocks up from Barney's (going east) on Santa Monica Blvd. is
Hugo's, a place that draws a surprising number of celebs, especially for breakfast and brunch.
The Seeing Stars guide is very handy for visitors who are trying to spot celebs. Here's their entry for Hugo's:
http://www.seeing-stars.com/Dine2/Hugos.shtml
I haven't been there in years, but near the Melrose Ave. entrance gates to the Paramount lot is Lucy's El Adobe:
http://www.seeing-stars.com/Dine2/Lucys.shtml