See if you can get the activities calendar for Grande Ocean in advance--there are heaps of things that the resort sponsors as well. We've been taking my daughter to HHI since she was about 4 months old, and has loved it at every age (she is now 12.) Indoor and outdoor pools, activities, etc. (She learned to swim in the hot tubs at GO.) She went back one time at about age 7 and insisted GO had gotten a different human chess set--because the pieces were way smaller! (It took her a few years for her to believe what we told her--that *she* had gotten bigger so that the pieces were no longer as big as she was, LOL!)
They have a new stargazing activity at GO--be sure to check it out. They've found a local with some pretty amazing equipment and you'll see some fantastic sights right there at the dunes!
I would definitely check out the schedule for the Coastal Discovery Museum as well. There is a farmer's market that sets up out there--and one of the vendors brings baby goats! Who doesn't love holding a baby goat?!?
https://www.coastaldiscovery.org/
If there is truly a rainy day (and they are done with the indoor pool,) there is also a good Children's Museum on the island as well.
https://www.thesandbox.org/
And definitely check out Marty's restaurant thread for HHI to find good child-friendly options.

(When we went to
Gusto on July 4, there were children of all ages at every table but one.)
Fish Camp also has lots of activities and places to explore while (if) you have to wait for a table: cornhole, etc.