The cheapest island for accommodations is the Big Island.

We've stayed for a total of 5 weeks at the Hilton Kingsland since 2021. We've spent lots of time since then on other islands as well. While it's not perfect, it is a great all around experience.

Kingsland is a timeshare (don't worry when you book through hilton there are no requirements for sales pitches). I've gotten in cheaper by exchanging in (typically from $600-$1400 per week), but the rack rates are really quite good.

As an example:
You can get a 2 bedroom for a shade less than $3000 for a week. For your group size, you would probably be fine with 2 2 bedrooms (could be a little tight, but it sounds like kids skew young).

Here's what we like about Kingsland:

1) It feels like a resort. The "Super Pool" is a lot of fun for kids with waterslides, hot tubs, and some really cool pool design. There are plenty of grills and some nice grass areas for throwing around a football or doing other family activities.
2) You get access to the Hilton Waikoloa Village. The pool areas there are cool, and our kids love going into the lagoon to swim with turtles. The snorkeling is honestly not good, but you are guaranteed to see dozens of turtles.
3) We like to plan an activity each day and then we'd end up swimming at the pool to fill in the rest of the day.
4) A shuttle will take you to A Bay...it isn't the best beach, but it's solid.
5) There are on site activities that our kids love.
6) The Big Island is great. It isn't for everyone (you have to drive quite a bit), but our kids love it there

For two rooms that puts you at about $12000 which leaves you about $465 per person for activities / food / gas. For a week, or a shade longer, I think this is doable (and my family would come in less than this).

We typically spend about $300 per week in groceries. I'll use the grills on site to do burgers, or grill up some freshly caught fish (I don't like it, but my wife does), and will eat out just a few times during the week (pick up huli huli chicken somewhere, fill up on shave ice, a well received local place, or if we just need a quick bite we'll go to a fast food joint).

I find that outside of the tours that can't be done anywhere else, our activities end up being really cheap or free (snorkeling, beach time, national park)

Some highlights for the Big Island:

- The volcano. It's better when there is lava flow, but it's still a good visit when it isn't erupting.
- Manta Ray Tour (for the older kids / more adventurous).
- Snorkeling is top notch on the Big Island. It's hard to beat two step.
- You will have to drive to the beach, but there are some great beaches a short drive from Kingsland (the beach at the Mauna Kea resort might be my favorite beach in Hawaii). Hapuna Beach is another solid option
- Unless there are accidents, traffic is pretty non existent.

A few other things we've really liked:
- Hawaii Tropical Botanical Garden.
- Polulu Valley hike
- Kahalu'u Beach Park (Great snorkeling and really easy for even young kids during the summer)
- Kayak to the Cook Monument (or tour). The snorkeling is excellent and Kealakekua Bay is beautiful. I've gone when there were dozens of dolphins swimming within feet of my kayak. The rules for landing on the cook monument have changed several times since we started going so make sure to research, but older kids and adults would like this A LOT.

Just as some general advice:

Your accommodations are going to be the most expensive thing. I'd choose a place that you are happy to hang out at because you have such a wide range of kids, it's great to have a hangout spot and smaller groups can do more adventurous things that appeal to them.