You know your child best. In general, even if well behaved I'm not sure I'd bring a 6-year old to some of places I'll mention below, but I'll leave it to you to decide. I've also tried to point out some places that I'd not have as much hesitation about bringing a young child.
There's no shortage of good restaurants in the area. The more well known ones would include Wolfgang Puck's Cut Steakhouse, and The BLVD, both in the Beverly Wilshire Hotel (famous for its use in the movie Pretty Woman), and Wolfgang Puck's Spago. Others of note include Bouchon, Il Pastaio, Crustacean, The Grill On The Alley and Mastro's. And, there's Belvedere in your hotel which is very good. For places that would tend to be more kid-friendly but generally not as fancy, in that area I would look at Il Fornaio, Cheesecake Factory, Jack 'n Jill of Beverly Hills, or The Farm of Beverly Hills.
A bit further away...
The Ivy on Robertson Blvd. - doing lunch at the Ivy is an almost cliche` way to try and spot celebrities.
Craft is one of the better restaurants in LA and is in the adjacent Century City area.
An assortment of restaurants on Beverly Drive south of the "golden triangle" area, such as Ruth's Chris, Chin Chin, and others.
And an assortment of restaurants along "restaurant row" on La Cienega Blvd., plus the surrounding streets - this is all at the east end of Beverly Hills or just outside of Beverly Hills. Restaurants here would include Lawry's The Prime Rib, Fogo de Chao, Koi, Ago, Fig & Olive, and others. Options that are more kid-friendly would include PF Chang's or the Grande Lux Cafe.
Edited: 10:19 am, April 15, 2013