Besides Deep Blue, the other operator that works out of Lobster Pot in Georgetown is Wall to Wall - I recommend them regularly for cruise divers. They leave from a dock just north of the cruise port so it minimizes the chance of missing the ship. And they work within cruise timeframes. The Lobster Pot itself recently closed (burned?) but the dive operations are still based there.
The Lobster Pot Dive Center Grand Cayman - one of the few dive centers with our own dock located on the west side, with Cheeseburger Reef, just a short swim from our shore. It's also walking distance from the cruise port.
Plan on a small delay as you're tendered in to Georgetown. Yours may be one of 5-6 ships hovering offshore. IMO it's a good time to be somewhere else, downtown is pretty small.
Or if you're there all day, you could get on an early afternoon
dive boat to Stingray City after your morning dives - ours left at 1PM - it's about a 20min. van ride over, maybe 20+? mins. out and a 45min. dive. It's only 15' deep so your son would have no issues - and a lot of fun. Just about everybody does it, likely including both Deep Blue and Wall to Wall so maybe go with them.
I don't see why your son couldn't do the Kittiwake also as that's a small group escorted dive (as are all dives on GC) and an easily penetrable wreck. CITA does have some restrictions in place for decks/penetration based on certification level but I don't recall seeing anything about age.
The Official Website for the Kittiwake Shipwreck - Home
You can also cab 5 mins. south to Sunset House and spend the day there. The Mermaid dive and a small wreck just offshore - a lot of it is 40-50' the base of the Mermaid is at 55' - and they have the salt water pool for easier entries. Also the Cathy Church photo gallery/school on site. My Bar there is good for lunch.
Cayman Islands Scuba Diving Resort
The best accessible shore dive is probably Turtle Reef up in West Bay. Under 30mins. by cab/car depending on the congestion near the Ritz. Sundivers is on-site there and Macabuca Tiki Bar for lunch. It's a mini-wall 40 yds. out that bottoms into a sand flat at 50ish feet. The wall all the way down is covered in stuff - there's usually hundreds of Tarpon hovering in the shade to the south.
Cracked Conch Grand Cayman Ocean Front Seafood Restaurant and Tiki Bar Shorediving