Ben: The two best SCUBA places on the Big Island:
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At the Captain Cook Monument, which is tough to get to. You walk down a long, steep and often overgrown trail, or you take a kayak and cut across the bay. TONS of fish, some turtles and dolphins.
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“Two Step” (named for a pair of underwater shelves near the shore). This is right next to the Place of Refuge. Lots of fish (not as much as #1), but LOTS of turtles. Also, several underwater arches 100 meters or so offshore.
If you’ve ever been to these (I was there in December 2009), you will turn up your nose with haughty disdain at anyone who raves about the Hanaumau Bay Preserve on O’ahu. The coral there is mostly dead, and the only reason fish even come there anymore is because the Park Rangers toss out pellets of fish food in the morning.
To the OP: I’ve been to the Hawaiian Islands on four separate occasions, most recently to Maui and O’ahu just last month. Try to get to an island other than O’ahu, but only if you can set aside at least four days to spend there. Inter-island flights are relatively cheap.
On O’ahu, by all means go to Lanikai Beach, just down the shore (south) from Kailua Beach. If you can, take a kayak out to the 'Mokes (a pair of small islands about 300 meters offshore). People are only allowed on the one to the north (left); the other one is an off-limits bird sanctuary. If you/your family is adventurous, you should hike around to the back of the island, where there is a small and rugged bay/inlet. Look at, but do not imitate, the crazy people who jump off a spot on the cliff into the bay. Some people have died doing so.
Snorkeling between Lanikai Beach and the 'Mokes is pretty good, too. Some turtles there.
If it’s a rainy day, go to the Bishop Museum. Stop at the Pali Lookout on a sunny day. Skip the Dole Plantation. Get to the Windward side; it’s less developed, more rugged, and very “native” – and as such it will give you a different perspective on some of the politics/economics brewing under the surface in the islands. (Do some reading up before you go to give you perspective.)
Get to Pearl Harbor EARLY in the morning. The boat ferrying people out to the Arizona is not large, and spaces fill up fast on a first-come, first-served basis. Get there after after 10 a.m. or so, after the tour buses roll in, and you may not get a ticket until a 4 p.m. boat ride – or you may not get one at all.
A boat trip to Sunken Island in Kaneohe Bay is fun, too. It’s a sunken sand bar that sometimes pokes above the water a bit.
As for accommodations, I recommend staying with my sister at her condo in Kailua! 
(But seriously, others’ advice here on places to stay are very good.)
If you want to stay in downtown Honolulu, that’s fine and you can use that as a base if you want to explore the rest of the island. The island’s not that big, and day trips are very possible. But just so you are aware, if you don’t time your excursions well, plenty of your time will be wasted in traffic jams. As the hotel/resort staff for tips to avoid traffic.
Have fun!
It’s great if you want to see a variety of Polynesian dress and dance, but I imagine someone who’s not from Hawaii would start to blur them together. Hell, I’m from Hawaii and I blurred them together. Still, it’s an entertaining show. Probably not worth the long drive (though there’s great scenery along the way).
I always enjoyed getting away from the south shore (especially when I lived in Waikiki), taking the motorcycle on less congested roads, to enjoy the quiet of the North Shore during the slow summer season. I especially liked driving through (by?) Kaaawa…always traveling counter-clockwise around the island, so as to see the Windward side during daylight, with the added benefit of not going over the Pali or H3 at night (cold and windy on a bike).