LA For Lunch
I've only been to Doug Arango once, for dinner, and I thought it was fine and service was fine, but nothing was great or special that has motivated me to return, even the french fries that I was told were amazing were just odd to me. But it's a nice atmosphere and your kids should be fine there.
Do you need to be in the West Hollywood area for lunch or can you travel outside of it at all?
In Hollywood there is 25 Degrees, which has really good burgers. You choose the beef, bun, and toppings, and they have tons of different kinds of cheese. It's a pretty hip/trendy place, might be fun for the kids, and it's across the street from the Mann's Chinese Theater and the Hollywood/Highland complex.
Also, in West Hollywood there is Vito's Pizza which is great east-coast style pizza, if that's your thing. Super casual though.
25 Degrees Restaurant
7000 Hollywood Blvd, Los Angeles, CA 90028
846 N La Cienega Blvd, Los Angeles, CA 90069
Going a little bit further south, the 3rd Street Farmer's Market would be a great destination for lunch, especially for a group with kids as the great variety allows everyone in the group to go where they want and then just meet up at a table.
Orris is a really nice Japanese-French fusion that serves "small plates" akin to Spanish tapas. Not sure if this is the type of place to be taking kids, though, as I know some adults who would not like the cuisine. That being said, I like it a lot, especially if I'm in one of those moods where I want my palate to be intrigued.