The only free campground around the Lone Pine area used to be Tuttle Creek Campground, off the Horseshoe Meadows road. That campground is a little east off the Horseshoe meadows road, not long after you leave the Whitney Portal road. It was free but asked for a small donation. I haven't stayed there this year.
You can camp for free in the Alabama Hill, but fires are not allowed.
Horseshoe Meadows campground is $6 like Passingthru said......and well worth the money.