A top spot is Mosquito Flat (don't let the name deter you!). It's easy to access off Highway 395 and the trail takes you along a string of scenic lakes and mountains as far as you want to hike over a weekend out and back - perfect for a family trip. You'll be in bear country, so be prepared!
Here is a link to the Mosquito Flat backpacker's campground at the Little Lakes Valley Trailhead, where you'll find additional information on the surrounding area in the Inyo National Forest including ranger station contacts, permitting requirements, and helpful details. Happy hiking!