Be sure to bring:

Flashlight and maybe some kind of light for your campsite. Otherwise, you may be stumbling in the dark. Some people bring Christmas tree lights to light their site!

Matches or grill lighters for your campfire

You may want to buy some firewood off site somewhere

a hat for shading your head/face
some people use sunscreen
Aloe or some other lotion for sunburn

Bathing suit
Towel

There is a full bathhouse across from the campsites, but you might want to bring a roll or 2 of toilet paper, in case it doesn't stay stocked.

All the usual toiletries to use in the bathhouse.

a folding camp chair, otherwise plan to sit on the ground a lot of the time.

Remember that it'll get pretty cool at night here, even in June. To keep warm when sleeping, it's really a good idea to have a pad or air mattress to keep you up off the cold ground. You'll probably need blankets in addition to a sleeping bag, too. Better to have plenty to stay warm with than to have a chilly night.