I assume you'll be traveling around on public transport. If that's the case, your town on the AC needs to be one that is good for the buses and ferries. (FYI, both revert to off-season frequency in the second half of the month.) And town with both bus and ferry service will be touristy. I prefer Amalfi, but most here swoon over Positano. You can look for hotels somewhat removed from the busiest areas, but that will be mean longer walks (and/or many stairs) to the bus stop or ferry dock. If you're looking for a quad room as opposed to two double rooms, that might limit your options but I don't know for sure as I've never needed a quad room. I wouldn't stay in more than one hotel on the AC. I don't know your kids, but teens I know would probably find 2 full days (3 nights) on the AC to be enough. If you were thinking about a hotel with a pool, you probably won't find a pool that's heated in October.

On Capri, anything away from the main piazza in Capri town and away from the ferry dock area will be less crowded. I like Anacapri, and I think your kids would enjoy taking the chair lift out of the town to the top of Mt. Solaro. I wouldn't schlep 4 people to Capri with luggage for just a single night.

I never know how people will react to Naples. I like the city, but it's the opposite atmosphere of Capri and the AC towns. I wouldn't miss either Pompeii or the archeology museum in Naples, so I would stay at least 2 nights but 3 would be better. Depending on interest and teen tolerance, you could spend many hours wandering around Pompeii. The excavation is pretty big and IMO very interesting. Hire a guide for part of the time if you think that would increase your kids' attention.

If you want hotel recommendations, you need to state a budget and indicate the type of room/rooms you'd want.

You may consider basing 2 nights in Salerno - more of a regular larger town with non-touristed areas and the chance to day trip by bus or train the short distance to Paestum and its great collection of Greek temples. Also on mainline train line to Naples and Rome.

https://www.google.com/search?q=paes...iw=944&bih=526