Hi!
I am going to the island of Oahu in Hawaii this August.
Can you help me to find the name of the cheapest room for the nights: guest-house and camping as well. I heard that it is possible to find rooms at 50 USD / night but I didn't find it on the internet.
Thanks in advance
Bertrand.

Hostel International near Honolulu University, or Waikiki Beachside Hostel.

There are three hostels in Waikiki about 1/2 blocks from the zoo and the beach. they are all the same owners, I believe. One is "The Beach Waikiki boutique Hostel. Dorm rooms are about $40 per night. Check Hostelworld.com for more info. You can also try AirB&B. Depending on when you arrive in August, some college students may have rooms available if their roommates went home for the summer.

$50/night is hostel territory.
To camp you need permits and you would have to drive or take a taxi to the campgrounds since you cannot take luggage on the Bus other than a small daypack.

Where on Oahu? Do you care where on the island you stay?

How are you going to get equipment for camping out?