Oahu does not have any National Parks you can camp in, but there are county and state parks that will let you pitch a tent. Remember the general Hawaii rules that the windward (east-facing) side of any island will be rainier than the leeward (west-facing) and the south side may be windier.

Also remember that although crime is relatively low in Hawaii, it does exist, so stay together, or near other people and keep an eye on your stuff, just to be safe. Don't leave anything valuable in your car, as there are some vehicle break-ins in quiet areas, just like any other tourist area.

Oahu State Parks and Reserves that allow Camping

Most state parks have a campsite fee of $5.00 or $10.00, and you need a permit (free). Also, there is no camping on Wednesday or Thursday night in most of these sites. Get more info and download a permit application at the DLNR website for the following campsites.

• Malaekahana Beach Park
  Located on the windward side above Laie. Excellent beach, with available beach houses and campsites. Has restrooms and showers. - Highly Recommended!
  Malaekahana Campgrounds info and pictures --- Extensive Personal Article about Camping in Malaekahana

• Sand Island State Recreation Area picture
  Located on Sand Island in Honolulu in the airport's flight path, so it could get a little noisy. Small beach with restrooms and showers. Fantastic views of Honolulu and the shoreline, and a great base camp for exploring Honolulu. (open for camping on the weekend only)

• Kahana Valley State Park
  Located on the windward side, near Kaaawa. Lots of hiking, and a popular bodysurfing beach, plus a fishpond. Has restrooms and showers. Beware of the mosquitos if hiking in this area. Some Kahana Valley pictures and a Hiking travelogue

• Keaiwa Heiau State Recreation Area
  Located above Aiea, on the South side, at the base of the mountain. No beach (not near the ocean), forest area with hiking and remains of a heiau (hawaiian place of worship). Has showers and restroom. Located at about 1000 feet elevation, so it can get cooler than by the ocean. Keaiwa Heiau Pictures

Honolulu (Oahu) County Parks that Allow Camping

You need a permit again, but most are free. Again, no camping on Wednesday and Thursday nights. There are several county parks with camping allowed, so I am just going to cover the ones I especially recommend for safety and amenities. You can see the full list here.

• Bellows Field Beach Park
  Located on the windward side, above Waimanalo. Excellent sandy beach. Restrooms, showers, and lifeguard.

• Kualoa "B" Regional Park
  Located in Kaneohe Bay on the windward side of the island. Long beach with lifeguard, restrooms, and showers. More Kualoa Park Info

• Waimanalo Bay Beach Park and Waimanalo Beach Park
  Both located on the windward side in Waimanalo, and both are gorgeous with lifeguards, showers, and restrooms. Good for families and kids.
  

I would not suggest camping in any area near Waianae, although there are a lot of camping sites here, and the weather is usually hot and dry *and* the beaches are just gorgeous - Waianae is one of the "tougher" areas of the island and there tends to be homeless people attempting to live on the beach. You probably won't have a problem - but hey, who needs to worry about that on their vacation, right?