I think it's be pretty hard to do Hawaii with a family and without a car. Try Kayak.com for cheap rentals. (They have an Android and iOS app as well). Last time I was in Honolulu they had some well priced pickup trucks listed. Can't remember the rental company though. The only issue is you have to catch a taxi to the pickup location as they are considered truck rentals. Picking up from the airport adds quite a bit of money.
If you need gear storage at Honolulu airport try these guys:
www.alohaluggagedeliveryandstorage.com/The owner is called Dr Bob. They have a warehouse near the airport and storage is pretty cheap. I used them twice and both times the service was fault less. The first time they stored two double board bags for 8 days while we did some island hopping. It was cheaper to store with them and do a quick return flight from Maui to pick up the gear than dragging it around with us. There is no luggage storage at the airport itself.
If you want a to do a fun thing with your kid try the Disney resort out on the West Coast of Oahu. It's a bit touristy but they do a disney character buffet breakfast and you can afterwards stay in the resort for the day and use their facilities. We did it with a 1 1/2 and 3 year old and they thought it was the best thing in the world. And besides who wouldn't love Mickey Mouse shaped pan cakes.