Buckifan...
As XLT800 suggests... doing things on your own or tagging along with another family to split a private taxi van etc (for example ~ to drive privately around the island for the day) is an easy undertaking.... and money-saving while expanding and enhancing the experience.
I can recall in earlier days when we met folks on the ship who explained how easy it was to do things on your own when we had just returned from a pricey cruise ship excursion... and after several cruises we started trying a few things on our own... then as we advanced in courage we met folks on the ship with a quiet consideration for the potential of sharing a few activities (or for finding another couple who knew more about smart things to do for maximum fun with extreme value in port. So... before the advent of the internet we would have to chance it to meet folks or we would simple do things alone. As once on Jamiaca we chartered an entire bus with about 20 seats for a private island tour and we could have easily added one to perhaps 4 more families for the same (or slightly more) cost!
So... I know your kids are supposed to be surprised... so I was thinking to ask you under these secretive circumstances what do you think your kids might like. So... let me suggest a couple of basic considerations. Which of the following would you say sounds appealing...
1) staying on the ship to enjoy easy (constant) food and activities like the kids' clubs, internet, movies, pools, or simply loitering in the cabin and exploring the ship. I did exactly this in St Thomas with our 2 kids while my spouse went ashore to power shop without any restrictions on her own.
2) going to one of two types of beach for half a day or longer... type 1 being moderately commercialized with conveniences like cold drinks, hot food, bathrooms, plastic lounge chairs with umbrellas for a fee ($10 typically) and probably a few local folks wandering around selling things.... or type 2) under-developed, semi-private with minimal resources but more privacy, and much more of a spot locals would frequent.
3) Snorkling or scuba diving
4) Aerial tours... either fixed wing or rotary.
5) Historical guided tours ... or quasi-introductory from a bus type exploration
6) water craft adventures - including sailing, or cruising, or renting a small craft like a boat, or even sport fishing as well
7) shopping around town. An easy and potentially inexpensive thing IF you don't buy much. On the other hand this sort of thing could bust your bank if you're bent on acquiring another pair of his and hers platinum Rolexes OR Patek Philippes ... with diamonds.
8) site-seeing which is more a "lets see what we have not seen since we're here" sort of stuff... which is a sort of "happy-go-lucky" sort or approach. My spouse and I are often of this attitude since we are not put off if our exploration yeilds more or less discovery.
9) exotic site-seeing ... which might encompass the pricey stuff like private Dolphine encounters or renting a real 2 person submarine guided by a scuba diver chaufferring you from outside. And... I guess aircraft charters would have counted here but it has it's own line above since it is not all that expensive if you're good at bargain hunting.
10) Resort hopping ashore. This is like stepping off the ship to lounge at a decent enclave of comfort, convenience and safety.
So... beyond all this is a virtually limitless expanse of possible undertakings which are limited by your imagination and budget. But these 10 catagories probably cover a lot of the main ideas and concept catagories.
So... Buckifan.... what two or three things sound appealing to you and your family considering their tolerances and likely preferences? Am I putting you "on the spot" If so I apolagize (misspelled?) and please free to demure or deflect the direction of discussion.

As I'm just being neighborly and a motor mouth with a keyboard!
OK... TTFN....

Tom & Alicia