I love Columbia! I think it's an ideal place to make history come to life for kids--not too remote, no roughing it unless everyone wants to, and small, interesting doses of living history that don't wear the kids out.
This is on the way to Yosemite, if you had thoughts about going there (a bit late now to look for accommodations, however). If you're not in a big rush, I'd suggest taking a scenic route; one of them is Hwy 140 from Merced and then up Hwy 49, the "Mother Lode Highway."
The town of Sonora that Hilaron mentioned is one of the major Mother Lode communities. It can get busy and congested on weekends, but it has many fun shops and restaurants. Also see nearby smaller towns, such as Angels Camp (the Mark Twain "frog" town), Murphys (lots of shops and restaurants), and Chinese Camp (almost deserted now, but many old buildings and remainders of the Chinese miners who settled it). Every town has a historic cemetery, and these can be fascinating for kids.
The nearby town of Jamestown has a historic railroad that your family might enjoy.
http://www.csrmf.org/railtown/
Sonora has the most lodgings, but I love staying right in Columbia. There are two historic hotels, the Fallon House and the Columbia Hotel. The Columbia is on Main Street with a balcony overlooking all the action. If your budget is a little tighter (or you'd like a pool), the Columbia Inn is across the road from the state park entrance and just a stroll away. (Cars stay outside the historic part of town, which helps maintain the historic atmosphere). Be sure to go up the hill to the historic school and show your kids the "facilities." The cemetery is next to the school.
One of the little pleasures I enjoy is walking out in the town at night after everything closes, and imagining what it was like 150 years ago when horse thieves, madams, & stagecoach robbers hung out on the streets! Today it is perfectly safe to walk around anytime and is is altogether a great experience for families.