Just returned from a week long sailing trip in the Aeolian islands. For those who have already done the grand tour, I highly recommend this as an alternative way to see this beautiful corner of Italy.
The best meals are to be had on Lipari and Salina. In Lipari, the best known restaurant is Fillipino. Because it is so well known, I thought it might be a tourist trap and made a reservation at another place. However, the locals we spoke with insisted so emphatically that Fillipino was the best, that we went there, and were not disappointed. Fabulous food, and service like in a three star restaurant.
On Salina, the best place is Porto Bello. In Sicily, you eat a lot of raw fish and seafood, almost like Japan, and we had a lot of it in Porto Bello, all of it delicious.
We also ate at the Hotel Signum on the other side of the island. Food maybe not quite as good as Porto Bello, but a lovely ambiance, and also recommended.
On the tiny island of Panarea, it is the "in" thing to eat on the terrace of the Raya hotel. We had lunch here, and while the food is not amazing, it is quite good, and the view of Stromboli and the surrounding islets makes it more than worthwhile. It must be one of the world's most gorgeous views.