Greek Island

If you are willing to travel a bit further than Hydra (e.g. a flight or long ferry ride) I would suggest Crete- it does have amazing beaches, but even if you aren't interested in those it has wonderful hiking (Samaria Gorge) and other tourist spots (the Venetian old town of Chania, the old town at Rethymno, ruins at Knossos, and the amazing archeological museum). The bus network is very good so you don't need a car.

Another good option I would consider is Rhodes- the old town is one of the best in Europe and I believe they also have good hiking (tho I when I went I didn't do any so can't speak to experience). They also have a great bus network. This one would also require a flight or very long ferry.

I don't have as much experience with islands close to the mainland beyond Hydra so will let others suggest options that are a short ferry ride.

With just 2-3 days I would pick one of the Saronic Islands as they are the closest to the mainland.

One of the better ones is Aegina which has everything one would want in an island including one of the best preserved temples in Greece, an abandoned Byzantine Village (The Mystras of the Aegean) in the hills, seaside villages, and is known as the Pistachio Capital of Greece.

Many people suggest Hydra which is indeed lovely but it's mostly hanging out on the waterfront people watching but does have a lovely village cascading down the hillside. Most people don't go up they as they lanes and steps are steep but it's much quieter and more "Greek" than the waterfront.

One island I would recommend is Poros which has more to offer than Hydra but a little less than Aegina. Long waterfront with lots of options for shops, markets and tavernas but for me the best part is uphill away from the waterfront where it's classic Cyclades Architecture with white washed houses and colorful trim, narrow lanes and nice views.

Any of those would work for the time you have.