Within just 20 miles of the Organ Pipe National Monument, Gunsight Wash allows dispersed camping up to 14 days. There are enough flat parking areas scattered throughout the BLM area to accommodate hundreds of rigs. You should be able to find a quiet spot, though it's unlikely that you'll have the place all to yourself. 

As with most dispersed camping areas managed by the BLM, there are no services for bathroom, water, or trash. There is a day-use rest area across the highway, but please do not dump your trash there (AZDOT will charge the BLM to dispose of this trash, and that will mean that the BLM will have to start charging us -- so be a good neighbor and dispose of your stuff in the nearby town of Why). 

The main road between Why and Sonoyta (Mexico) is very busy during the days. 

However, the border crossing is closed at night so being parked near the pavement isn't going to keep you awake all night. However, we found that getting further away made the hanging out around camp more pleasant.

There are many hiking and biking opportunities available just a short drive away into the National Monument. The Visitor's Center is also worth a visit, and the NPS campground is super cute and offers all the usual amenities of developed campground in the NPS. The town of Why is within 2 miles of the Gunsight Wash turn off, which offers a gas station and convenience store. Further up the road lies Ajo, AZ, which has more in the way of restaurants and grocery stores.