Depending on the time of year:
Mt Bigelow (~7500-8000')loop road has dozens of campsites. These are primitive and walk-in, but great sites. No fee, no reservations. (Ideal for climbing upper mountain, Summit Crags, Druid, etc.)
Showers Point (~7500') is a group campground you can make reservations. (Ideal for climbing upper mountain, Summit Crags, Druid, etc.)
Rose Canyon Lake (~7000') is a nice area with dozens of paid campsites and more developed facilities, where you can make reservations. Definitely the most developed camping on the mountain as far as facilities, if you care. Also, more crowds, so it's a trade-off. (Convenient to Munchkinland, Troll Wall, Willow Canyon, etc)
All the above areas will be pretty cold at night due to high elevation.
Prison Camp (~5000'), mentioned above, is now called Gordon Hirabayashi. There is no 'established campsites', but you can camp there free.
Lower down is the General Hitchcock campground (~4500), and finally the Bear Canyon campgrounds, both good options and significantly warmer and more sheltered than any of the above.
There are also dozens of dirt roads all over the mountain where good camping can be found. I'm not going to divulge my favorites, but with a little looking around in the daylight, you should find plenty of options.
Hope that helps.