We have been going to Greece every October for a number of years and believe you would love Naxos. It's one of the best Greek Islands with a wonderful combination of Old & New World Greece. I recommend staying in St. George Beach, just a short walk to the waterfront and Old Town. If you don't feel like walking too far there is a small square nearby with markets, shops and tavernas.

Spectacular mountain scenery, archeological sites, centuries old Byzantine churches, lovely interior villages, beaches, great food, etc.

There are a number of Greek Islands that would suit your family including Sifnos, Milos, Tinos, Amorgos, Astypalea and several others. Some have airports which make it very easy to get to.

Santorini is one of the most heavily touristed islands in Greece. Granted the scenery is spectacular but the crowds, costs and touristed-oriented activities could make give your impression of a Greek Island something that many consider not really "Greek".

As far as airfares, right now there are very low from Boston which we also fly from.

You can do a search on Trip Advisor, Sky Scanner or several other third party sites and if you find something that fits your budget and needs I would recommend going directly to the airlines' website rather than using a third-party. Some of them are not very reliable.