Mt. Lemmon has a lifetime of climbing on it...but be prepared for guidebook maps not to match up with what's actually up there. A few fires have done some major re-landscaping. Also, be prepared for some chossy granite. Also near Tucson is Cochise Stronghold...very cool climbing there.
I'd agree with Sarah, in that you probably won't need to drive 4-5 hours anywhere (Phoenix is 1.5 hours away). However, if you are willing to drive that far, that opens up a lot:
Queen Creek Canyon (pretty popular volcanic sport climbing)
McDowell's (trad granite...mostly old school ratings)
Superstition Mountains (lots of hiking...would recommend against climbing, unless you love sketch pro)
Jack's Canyon (popular limestone sport climbing, soft ratings)
Hell, you could probably hit the Flagstaff area as well. Google maps says it's 4.5 hours, but throw in traffic and it's probably a little over 5.
There's a pretty decent book, http://www.amazon.com/Arizona-Summits-South-Mountains-Points/dp/0966491955/ref=sr_1_1?ie=UTF8&s=books&qid=1287682771&sr=8-1
. Goes over everything from driving to roped climbing.
Feel free to PM me if you have any questions.