Re: Disneyland, you'll definitely want to stay down here, as Anaheim is about 30-45 minutes away from both LAX and Downtown LA, and that's without traffic.

Have you bought your plane tix yet? If not, consider doing an open jaw, flying into SNA and out of LAX. Disneyland is about 10-15 from SNA, and the Convention Center is about 15-20 minutes from LAX. Most of the rental biggies also will not charge you a one-way fee, as both airports use the same fleet of cars.

I don't know what your kid is into, but nearby the convention center:
--California Science Museum
--Natural History Museum
--USC Baseball (at home vs. Stanford)
--Dodgers (at home vs. the Brewers then the Cubs that week)