A lot of people on CH hate this place, but I eat hear all the time because I live in the neigborhood.... but Toast has a good 1/2 pound grilled burger cooked to order, accompanied really fresh lettuce, tomato, and red onion. Comes with your choice of lots of different sides (the sweet potato fries are great.) Nice outdoor seating options, too.
Toast Bakery Cafe
8221 W 3rd St, Los Angeles, CA 90048
Lucky Devil's might be a good choice as obsessed suggests.
The Counter might be fun for the kids, as they'll get to design their own burgers.
26 Beach in Venice
Boneyard Bistro apparently
I know you said no chains, but Houston's does good burgers.
I personally like 25 Degrees, but after dark it might be a little too "night-clubby" for your young ones.
I definitely recommend the Counter as well (technically it's a chain, but not at all in the same way as Burger King or McD's). Also Tops in east Pasadena.
25 Degrees Restaurant
7000 Hollywood Blvd, Los Angeles, CA 90028
The Original Tops
3838 E Colorado Blvd, Pasadena, CA 91107
2901 Ocean Park Blvd Ste 102, Santa Monica, CA 90405
Best place for you and your sons in WeHo is Carney's, on the north side of Sunset. Your sons will enjoy the train car, and you will appreciate the easy free parking. They are best known for their chili dogs, but the burgers and fries are better than average. I like them better than Astroburger or Okidog, and the clientele is more family-friendly as well.
I second the rec for Carney's. It's a place our 3-year old likes, and I think their hot dogs are some of the best in town.
Also, I can't recommend Tompkins Square enough. It was fabulous. It is very much a bar, but we were there with our son before 8, and there were other families with kids there. Best burger I've had in...I don't even know.
Welcome to Chowhound!
Yah, this topic never comes up :-).
With $4/gal gas biting us, it would be helpful to know how far you're willing to travel from WH and how upscale you like.
In the meantime, er, chew on these recent burger related topics:
My favorites are in here:
Go for JNJ!