One island for 6 nights

Stephenson - I would say add the island of Naxos in the mix as a place to to stay for another week. Naxos also very close to Mykonos so it would be convenient to travel to.

Crete - mostly Chania, a full day in Heraklion to visit the archaeology museum and ruins at Knossos if that interests you. Chania has a few sights, lots of restaurants, cafes and bars all along the water, interesting shops to browse (I'm not a shopper but I enjoyed the shops nonetheless). If you want beaches, Naxos is lovely.

What month?

Santorini and Mykonos are the two most promoted islands so I am happy you have an extra week to get a more authentic taste of a Greek Island. Naxos is on a direct ferry route and is between Mykonos and Santorini so it is a good choice.
We can probably give you better thoughts if you can tell us when you are traveling and what other interests do you have beyond chilling out?

Here is Naxos
Naxos Town http://www.flickr.com/photos/stanbr54/sets/72157632094558042/

Trip around Naxos

http://www.flickr.com/photos/stanbr54/sets/72157634605629689/
Naxos sunsets https://www.flickr.com/photos/stanbr54/albums/72157700728046481

For just a week I think Naxos has enough to offer you. You will have an opportunity to see most of the island.

Crete is huge and a week really does not work at all. However Chani has a good point. If you choose just one area to stay in you can enjoy that area and not attempt to jam too many far apart highlights into too short a time.

Chania is our favorite city in Greece. Stay in the old town harbour area and you will feel like you have stepped back about 600 years. Chania is also a good place to make a base for some of the sites in West Crete.
Chania https://www.flickr.com/photos/stanbr54/albums/72157632117917828

Gramovossa, Balos Elafonissi https://www.flickr.com/photos/stanbr54/albums/72157698932695812

Ancient Aptera, Melidoni Cave and village.

https://www.flickr.com/photos/stanbr54/albums/7215767211837963

Thanks for the responses. We are going ithe end of September. Our tour starts October 2.

Santorini & Mykonos are the two most heavily touristed and expensive of the Greek Islands. How "Greek" they are are open to debate.

If you want something less over-the-top here are islands I've been to and can recommend with photos.

Tinos (A Gem): https://tinyurl.com/y6w5ncb8

Amorgos (Just Lovely): https://tinyurl.com/y7bqwfma

Chios (One of the most under rated of the Greek Islands): https://tinyurl.com/ycgkrqza

Lesvos (Laid Back and Traditional, still working on photos):

Astypalea (One of the most beautiful Choras I've been to): https://tinyurl.com/y7mdg7rv

Ikaria (Something different with free-spirited locals with a bit of anarchy thrown in): https://tinyurl.com/y4e5dsj6

I've been to Sifnos, Milos, Crete (highly recommend), Naxos and Andros. Any of them would be enjoyable but the above islands are at the top of my favorite list.

Another option if you want a mainland experience is the Peloponnese filled with villages, mountains, beaches, archeological/historic sites and much more. A little different character than most of the islands but still very Greek.

Rhodes Island and spend time in Lindos and Rhodes Town and/or a resort between the two. You can also take a 50m ferry to Symi for the day which is considered to be the most beautiful of the Dodecanese. If you feel like taking a day trip to Turkey, you can visit the resort town of Marmaris.

Weather should be excellent then. Have a great time!

I would definitely recommend Amorgos and Astypalaia - both of them are considered 100% Greek, without the annoying crowds of Santorini and Mykonos, with authentic and traditional dishes and breathtaking views! Safe travels!

Thanks to everyone who replied. I guess I have more research to do to make a decision.

Yes, research is your best tool to make sure you are getting what YOU want and not what the tourist industry wants.

Don't let the glossy photos in High-End travel magazine sway you just like you don't want to stop in a taverna with glossy photos of food on billboards outside. That's a sure ploy to get the tourists in and it may not be as "Greek" as you think.

If you find an island that you may like come back and we'll try to help with advice.