Hi. Cayman is great, but I actually prefer SJU. Aside from this being a US territory not requiring passage through immigration/passport control, I really enjoy Puerto Rico as a place and people. San Juan is a vibrant city with a stunning old quarter, fascinating history, wonderful beaches, great hotels (in San Juan proper, I recommend the ritz Carlton or El San Juan for families) and unbelievably delicious food. Outside of San Juan, you have other choices that are more island paradise and less urban beachfront...I really enjoy Fajardo and the hotels there...the El Conquistador (a Waldorf-Astoria Collection Hotel and a part of Hilton) is a great hotel for families, although the service can be spotty at times and you need to choose your accommodations carefully to fully enjoy the location. It does, however, have its own private island, and is located within striking distance of the bioluminescent bay and El Junque rain forest. If you stay there, you MUST leave the property...at least for dinner or lunch...there are some outstanding choices, and I would argue that La Estacion (the station, located within 10 minutes of the property) is one of the best dining experiences I've had (and I have racked up 50+ Michelin stars in my travels.) If you wanna splurge on the hotel, within El Conquistador is the much more private and truly 5-star Las Casitas Resort (also Waldorf-Astoria Collection)...my preference, but comes at a price — especially during peak. All of these properties include dining, pools, water sports, beaches, spa and casino. El Conquistador/Las Casitas has a world-class golf course. Finally, you can do one of the new properties recently (past 3 years) developed on Vieques —.including the new W hotel.
As for AA...at present they have one non-stop—.leaves early morning and returns late afternoon so you get there early and leave late...flight time is roughly 5 hours on a 737-800. There are also a very large number of connecting flights as SJU is the AA hub for caribbean flights.
Realise this is less about AA and more about the destination...but then again, you asked

Hope this helps.