Swap Mykonos for something less-touristy but still lively, such as Milos, Naxos or Paros.

The second week of October will be off season for most islands,
so don't expect too much....
Islands like Milos will be dead, and the rest ( Paros, Santorini etc) will be preparing for the winter.
THe most active island will be Mykonos, but it won't be like it is during the summer season...
I wouldn't try to find Las Ramblas on any Aegean Island... since we are talking about small and often tiny islands compared to a metropolis like Barcelona.
All the known islands ( and most of the less known ones) have many cafes and bars and restaurants...but as the season ends any will be preparing to close if they are not already closed.

Islands with large permanent population are Syros, with magnificent Ermoupolis, the capital of the Cyclades, Rhodes and the cities of Crete. There you can find what you are looking for, and the weather especially on Crete will be much warmer, as it is the most southern point of Europe.

The weather can be warm, but not as in the summer.... On Crete you could even swim, while in most other areas the water won't be warm enough...

You could combine Athens with Nafplio, a beautiful city 200 km in the South of Athens, and full of shops, bars and restaurants and above all history...
Mycenae, Tiryns, Epidavros, Nemea are all nearby and ideal for day trips.

The problem with combining Santorini and Mykonos in the second week of October is lack of ferries between the two, as all direct connections cease by the 8th of the month. There are no direct flights between the two islands either, so that would mean a detour through Athens. You can still get between them via Paros, but even that isn't completely straight forward. If you start from Santorini, ferry timings would mean an overnight on Paros before proceeding to Mykonos on SeaJet 2 the next morning. Reverse the itinerary and you can take SeaJet 2 from Mykonos to Paros, then transfer to Blue Star, which leaves for Santorini about an hour later. Problem is, if SeaJet 2 is running late or cancelled you will miss the connection.

I like clausar's idea of Nafplio, which is on the mainland coast, and has lots to see and do. Likewise, Athens will be lively, and October a perfect time for sightseeing. If you still want to include Santorini in their itinerary they could fly there and back from Athens and spend a couple of nights on the island.

I don't understand why so many people think that in Greece only the islands are worth seeing. The mainland has more historical sights, a long coastline with beautiful beaches, and no worries about ferry timetables & possible cancellations. If they are up to renting a car, Nafplio is only 2-3 hours by road from Athens. If they don't want to drive, there is a good bus service, and if that doesn't appeal there are tour companies that will take them to Nafplio and the nearby sights. Chat, Key, and Go are Athens tour agencies offering packages that include hotel rooms and transportation.

Kos is a beautiful Island and if you stay in Kos town it would be very easy to get about, there is a little train that takes you to the Asklepion if they want any taste of history. There is the harbour and lots of lovely shops and cafes etc which seems to fit what you are after. We stayed in the Akteon Hotel which is right by the harbor. I think the weather will still be pleasant in October, maybe cardigans on a night?

Hope that's helpful