From our experience with younger kids, If 120 is open, a day trip up to Tuolumne Meadows... try going up Pothole Dome. It's not very high, nor difficult. A 6yr old can do it easily; he can imagine himself "rock climbing". (my son imagined it that way when he was 4). Then walk along the JMT towards the "twin bridges" in Lyell Canyon. Explore the river...
Bring bikes if you can, rent if you can't. A bike ride in the valley is great. Your best bet for saving money on food is to make it yourself. Groceries are available either at Curry (half dome) village or Yosemite Village (much better selection). Otherwise the pizza deck is good, and there is a food court at Yosemite Lodge.
If you don't mind sleeping in a tent and have warm sleeping bags, your son may enjoy being in his own tent rather than the tents at Curry/Half Dome Village (it may actually be quieter at night since the tent cabins at Curry are RIGHT NEXT to each other so you'll hear snoring, giggling etc).... just keep an eye on the weather as it can get below freezing at night. Don't forget to bring warm clothes...
General hint: Get your son his own headlamp, and something that glows in the dark...(check for Halloween stuff). Also, if you have a smartphone download a free star chart app for it...
Have fun!