Naples- A place to "get away from it all" for a few hours?

Here are two ideas:
-Santa Chiara Monastery cloister (beautiful majolica tiles)
-San Martino Monastery and Museum (wonderful views of entire Bay of Naples from here)

You don't say if you have daytrips planned to Pompeii, Herculaneum, and Sorrento, all about an hour by gritty commuter train? You don't need nine days just for Naples.

How about the Royal Palace of Caserta? I haven't been there, but there are some pastoral walks near the Gulf of Naples but out of town, I had some interest in the Cave of the Cumaean Sybill and Virgil's (supposed ... ) Tomb, but no time to go. It's a longer trip, but with your schedule I'd consider Paestum. It is much less visited than Herc and Pomp, and not like them at all. Hot and exposed, I imagine.

There are in fact five major Roman sites near Naples, and you can get a ticket for all five, including Pompeii and Herculaneum. You will be absolutely alone at the three less known (but still really important) ones. I'd put Villa Poppea third, after Pomp and Herc. But I haven't been to all five.

Edit: Don't overlook the important Farnese Marbles at the Archaeological Museum. If one of the teens is not interested in art and architecture, they might be interested in Lady (Emma) Hamilton, who may have been one of the most important British diplomats of the 18th Century, even though women had little professional status at the time. She also had a bit of ... er ... "Kardashian" in her youth.

Naples is intense. More than most cities and I do wonder why you are planning on such a long stay. Going out to the islands or hiking on the Amalfi coast would provide a nature break. My son had a hard time his first trip to Italy with all the crowds and the concrete. Fortunately we didn’t have a set in stone itinerary and headed to the country where we could ride bikes, hike, and walk on the beach.

We're definitely doing a lot of day trips-- Pompeii one day, Herculaneum another, Capri and also Amalfi. We decided to stay in Naples (as opposed to our original plan which was to also stay somewhere else) so that we're not rushing from place to place. My kids are pretty well-traveled and they did Rome OK several years ago.

I read once that if you can handle Rome, you can handle Naples. We did not feel OK bringing them to Naples when they were younger, but I feel they will do OK now, as long as there are lots of breaks with a bit of nature thrown in for good measure.

I do understand that the grittiness might be a bit for one of the kids, so 1. that's why we rented a pretty nice place :) and 2. definitely want to keep our itinerary relaxed, if possible.

Thanks for the suggestions-- I'll look them up!

Is there an event that your attending in Naples that requires you be there for so long? I agree on needing a break from the hustle and bustle of the city. We always do Big city, followed by a few days in a smaller slower town, then back to big city.
You are so close to the Amalfi Coast I would spend at least 3-4 days in Sorrento and hit Pompeii, The beaches of Amalfi, there are hikes you can do between towns. The You Tube video I am posting a link to is long but at about 53:01 he starts a walk between Amalfi and another town...I want to do this next time we go. The views are amazing, the slice of life you get to see on the walk is memorable and just looks so cool.
https://www.youtube.com/watch?v=EJ5Ju8-t3jk&t=341s

Get them to a beach, it should help

I second Agnes' suggestion of Santa Chiara Monastery cloister (beautiful majolica tiles). The tiles are beautiful, the garden tranquil. There is a small chapel museum. Three years ago there was excavation of Greek ruins going on beside the cloisters.

Recommend visiting the hilltop neighborhood of Vomero. The metro and several funiculars wil take you there. It is a quiet neighborhood with many trees. A particularly peaceful spot is Villa Floridiama, a beautiful park.

Caserta Palace and Gardens. 1/2 hour train ride.

https://www.reggiadicasertaunofficial.it/home-en/

Pozzuoli 15-30 minutes from Napoli by metro or train.
Flavian Amphitheater 3rd largest in Italy. Underground is well preserved.
Rione Terra old city of Pozzuoli has been undergoing reconstruction for about 15 years. Great English tours on weekends.

Baia …underwater archeological park viewed by glass bottom boat. Train from Napoli or Pozzuoli.

All these areas a much quieter than Napoli.

All are good for day trips

Gerri has some good ideas that may be of interest to teen boys.

Note that Sorrento is not really a beach town, it's built on a cliff. There are (mostly pebbled) beaches on the harder to get to Amalfi Coast, and in some of the towns "before" Sorrento. I don't know if the latter are walkable from the Circumvesuviana.

If I were a teenager. Or even now, to get away from Naples I would most certainly see about climbing Vesuvius.

Thanks, all. I think we will add Vesuvius to our trip to Herculaneum. I really like the idea of the glass bottom boat!