Mykonos is the Greek island that became a brand. Venetians built its windmills in the 16th century, pirates raided it for hundreds of years, and the Ottomans held it from 1537 until the Greek revolution. Then archaeologists started digging on neighboring Delos in 1873, the jet set arrived in the 1960s, and by the 2000s it was one of the most expensive islands in Greece. Underneath the glamour it's still a small place: about 86 square kilometers and roughly 10,700 residents, most of them in one whitewashed town.
That smallness is why three days works. You can't do Mykonos in one day, because the town alone deserves an evening and a morning, but you also don't need a week unless the beach is the point. In three days you can walk the whole of Chora, spend a morning on Delos, swim at a south-coast beach, and drive the quiet north and east where the island still looks like a Cycladic farm. It's also an island that's honest about what it is: loud, windy and not cheap, with the wind and the crowds both easing by October.
Day one is Mykonos Town, day two is Delos and the south-coast beaches, and day three is the rest of the island by scooter, taxi or car. Start on any day of the week: Delos is open every day except a handful of public holidays, and none of them fall in October. Greece itself is covered in more detail on the Greece page, and if the island is still undecided, our Mykonos vs Rhodes comparison helps.