A variation of our Bums and Bison trip would be excellent, if you don't mind a car shuttle.
Start at the Wapiti Lake TH, hike into Pelican Valley, over into the Lamar, over Specimen (would mandate a long day w/r/t campsite regs), and out the Black Canyon of the Yellowstone to Gardiner. Probably around 100 miles all told. The first half, which we did in 2.2 days, was just awesome. Great scenery, one other hiker, tons of wildlife.
Probably too much for a 10 year old. Hmm, any part of the above trip would be excellent. The area around Wapiti and Astringent Creek has tons of cool geothermal features. Maybe the Wapiti-Pelican-Lamar bit, but coming out over The Thunderer?
Have fun, it's an awesome place to go hiking!