Follow the road, fork right and head all the way in. You can get up on the ridges and have an excellent view for days. Plenty of space, roads are decent but get rough in parts. Ranger said no camping on the beach or waterfront so use your judgment.
Decent roads push out across and over the wash, giving you many great places to post up and spend some days or just drop a boat in. Coyotes sing at night, pretty harmless just be smart with your food and waste.
Getting up Thomas Mountain is good fun. This Campground is about 8mi up the ridge along a sometimes washed out road. The site itself was peaceful and quiet, but entirely left to run away. Not a lot of level ground.
Great place to stay free while exploring Joshua Tree. Southwest facing so you can spend a little more time in the park during winter months since it is a decent haul back down to the Cottonwoods.
Kelso Dunes is a beautiful spot to spend the day climbing or, if you're like me, three days watching the rain and frost roll in! The dispersed sites are spacious and fairly well kept up. Watch out for deeper sand that will sink you in good, they are easy to miss after dusk.
The dunes are stunning and easy enough to climb with any level of difficulty.
Nine Mile Coves is about 10mi south along Powerline Road directly inside the National Park Station near Cottonwood Cove. Free to camp, no reservations, but you will need to pay an entrance fee the park.
Sandy, exposed lot with a couple good sites. I'd say best for Vans or overlanding instead of trailer-towing. Lots of wind from all directions. Clean outhouse with trash bin inside.