I would highly recommend a cruise on the North Shore. The whales will be gone, but it still is fun.
My friend Robin is the captain of this one:
http://www.sailingcat.com/
You could also try skydiving at Skydive Hawaii:
http://www.skydivehawaii.com/
My friend Shaun is one of the skydivers.
Ok, now that I've pimped my friends businesses I'll give you some more places that I'm not affiliated with. I haven't read through the thread, so some of these might have been mentioned already.
Polynesian Cultural Center. The best luau, along with recreations of villages from the 7 polynesian cultures. Excellent show after the luau. That is where I take all my family when they come. One caveat: it is owned by the Mormon church, so no Mai Tais or other alcohol with the luau. If you want that try Paradise Cove luau. It isn't as good, but you can drink all you want.
Dolphin Excursions. In Waianae. They take you out in a rubber zodiac to see the dolphins, then to a good spot off of Makaha for snorkeling. A great experience.
Hike to the top of Diamond Head. It is a short but strenuous hike, with 250 stairs along the way, but well worth it. The view of Waikiki is excellent.
You mentioned the Arizona Memorial. You should get there early to get tickets, they go fast, but it is worth seeing. I work across the street from it so I see it every day.
Kailua Beach Park. The best beach on the island IMO.
North Shore. Lots to see there and my favorite part of the island I live near there. Haleiwa is an interesting town. Eat at Haleiwa Joe's while there. Just past Haleiwa is Laniakea Beach which ALWAYS has sea turtles laying on the beach. Just look for all the cars parked along the road, it is very popular. Waimea Beach is a nice beach to go to in the summer. In the winter the surf is too big for swimming, but you'll be there in June so it's ok. Right across the road from it is Waimea Falls Park. Run by the Audubon Society it costs I think $5 and you can walk up a scenic path to the falls. Past there is Sunset Beach, another really nice beach, and the home to the pipeline and the Triple Crown of Surfing.
If you have any questions at all PM me. And if you want to meet somewhere and have a beer while you are here I know all the best places....