Dollar rent a car has a few locations, one is just behind the Fort Derussy Park, right near where you are staying. Be aware that parking in Waikiki is expensive, ($20-25/night) so if you want to rent for a few days, plan to get up early one day, and drop it off later in the day you are fininshed. The car rental locations in Waikiki are NOT open late for drop off. Check what time they close when you rent it, otherwise you will end up paying an additional night of parking. Also, as reported here and other places, don't leave valuables in the car when you park at the North Shore.
The drive around Oahu is awsome, you will definetly enjoy it.