Arizona/New Mexico camping spots?

We live up in Seattle and would like to go down to visit our son in Arizona for Christmas. We’re a very outdoorsy family and would like to go camping somewhere in Az/Nm where we can be pretty secluded, do some hiking (5-10 miles), rafting, exploring (mines, ruins etc). Please no traditional/touristy suggestions. Thanks!

grand canyon would be perfect for ya but i know its not traditional