There are a lot of beautiful forested areas to camp up around Flagstaff in the Coconino and Kaibab National Forests.
I highly recommend camping up at the Mogollon Rim. The drive up through Pine/Strawberry is lovely.
I'm not sure if there's much for boondocking south rim of the Grand Canyon and you definitely don't want to go into the park there. Unless you like Disneyland nature. The canyon is f-ing amazing, but I recommend getting some north rim campering.
Perkinsville Road is a dirt road from just outside of Chino Valley over to the little used-to-be-mining-town-now-touristy Jerome. I've done it on my (dual-sport) motorcycle a number of times, but haven't in a four-wheel.
There are some nice places to camp off Williamson Valley road near Prescott. Also up there, camp up on Thumb Butte. The dirt road is traveled but beautiful views. When I was living in Prescott, I overnighted up there a couple of times just for a quick get-away. Same goes for Senator Highway.
East of the 17 is the Apache-Sitgreaves National Forest, somewhat north is Coconino.
It's a campground, but I highly recommend checking out the Navajo National Monument up off the 160 before Kayenta. When I was there, there were two others in the campground. Dunno what it's like this time of year, and honestly don't remember when I was there last year but it was lovely. There's a canyon and you can do an easy hike from the visitor's center down to see cliff dwellings. Good stuff.
If you continue up the 160 from there, you can go up to Utah and to Valley of the Gods. Camp down there, then drive up the Moki Dugway and camp at the top out at Muley Point. It gets *really* windy, but you can see Monument Valley from there. Good cell signal and gorgeous.
Near the (south rim are) I have Lipan Point marked on my google map, but have not been there.
If you go more east, Canyon de Chelly National Monument is amazing. I think it's mostly reservation land around there though, so boondocking is difficult. Cayuse and I went there this past October, and camped in the campground. Not ideal, but the monument was cool.
Petrified National Forest is also east-er.
White Mountains (east-er/Mogollon Rim stuff)
https://en.wikipedia.org/wiki/White_Mountains_(Arizona)
I'm working on a plan right now - heading towards Marble Canyon over by Page, then taking (dirt) House Rock Road north into Utah and
Cottonwood Canyon Roads.
more things on my map that I haven't yet explored: near Sedona - Boynton Canyon Trail and Cathedral Rock and I have Forest 525 Road marked too off of 89 near Sedona: zoom in and see boondocking sites
https://goo.gl/maps/BKc8JWYn61D2
Hope any of that helps! Have fun!