Here's what I found about water temp. in October. I looked up the water temp. for Lipari, since it's bigger and easier to find, but for a honeymoon, you're probably more likely to stay in Salina or Panarea (I've never been to Panarea). The average water temperature of the sea around Lipari island is 22 centigrade.

P.S. That's about 72 Fahrenheit. (Maybe a little cooler than you'd like for snorkeling, but definitely not cold for swimming.)

cmt: We go to Italy twice a year to visit family in southern Italy. I have been to Ischia and it was warm and then went to Amalfi in the first week of October when it is so cold that there was snow on the mountain going from Sorrento to Amalfi and on Vesuvius. The last two times we were in Sicily in the first week of October we had to wear jackets. It can be warmish but to plan an unforgettable honeymoon based on being able to sun and swim is not a good idea.

I am originally from the Long Island (not far from Jersey) area so I'm used to cold but from Naples to Sicily in the beginning of October is very iffy.

I think there can be snow on Vesuvius even when it's pretty hot down below. People have told me that they've gone swimming at the beach in Sicily after skiing on Vesuvius. I can't confirm whether it's true, but I think it's not unusual for it to be cold high up but warm at the coast.

The last time I was in Sicily it was in November. I thought it was pretty cold when I spent a few days in a little town in the Nebrodi Mts. in northern Sicily I was wearing a cotton top, with a polartec sweater over it, and a rain jacket over that to block the wind, and a light scarf. But 20 minutes away by car, a town on the coast where I was getting the train to Palermo felt like it must be in the upper 70s F, and I was hot in Palermo.

If swimming were not a primary purpose of a vacation, I think it might still be reasonable to plan an October vacation in the Aeolian islands, with a reasonable expectation of enjoying some swim time. But, if daily swimming is a must, and especially if hours of snorkeling are part of the plan, I agree that Italy is probably not a good choice, not even the Aeolian islands. 72 degrees F is nice for short swims, but a little chilly for spending hours at a time in the water as one might in the Caribbean.

Here is a related question, in one word, about sea swimming off the Italian coast/islands:

Jellyfish?????

Apparently there are jellyfish, from the Aeolian Islands website:

In the seas around Lipari are quite a few jelly fish. They are not big enough to do any serious harm, but they have very painful stings. I would strongly advise having a snorkel and mask if going into the water, as you can spot and swim away from them.

Also listen out to other people. Because if you hear someone shout "Medusa". You know there is a jelly fish close by.
If you catch a jellyfish in a bucket make sure you bury him on land so some one doesnt stand on it.