There isn't a great deal to do on Little Cayman if you're not diving. It's more of a "hang in the hammock with a good book" kind of place. There's a couple small museums that could kill an hour and you can watch the birds at the sanctuary. There's fishing trips. I like to take walks. Definitely need to be able to amuse yourself. You'll note I didn't mention shopping, but I suppose you could kill an hour in the gift shop if you try hard.
Do you have an idea where you want to stay? I've stayed at Little Cayman Beach Resort every time (the largest place though that's not saying much) and they have a few things to do, though I rarely see anyone doing them. There's a small fitness/game room with a pool table and book swap area. There are bikes to borrow, darts by the bar, a combo tennis/basketball court, hobiecats and kayaks for rent, volleyball net on the beach, maybe horseshoes. The pool and beach are pleasant. There's a tiny "spa" across from LCBR but I'm not sure if it's still really open, supposedly there is someone doing massages out of Southern Cross.
Swimming and snorkeling off the beach aren't actually such good things to do from LCBR as the water is very shallow and grassy - or I assume any of the other places in the sound like Southern Cross, Conch Club, The Club. I suppose you could if you really wanted to but I've never seen anyone try. Not sure about the others like Paradise Villas, Pirates Point, or Sam McCoys. I think Point of Sand at the far end of the island is supposed to be good snorkeling, , maybe Owen Island which is in the sound, and I think there are a couple other places but you have to get to them. It's possible to go on the dive boat to go snorkeling, for a price I'm sure, and assuming they go to a site that starts shallow enough.