I absolutely loved Alonissos mentioned upthread, and it is quiet and traditional. Might be too quiet perhaps. Skopelos is also worth a look, it’s also a boat ride from Skiathos, it’s the “Mamma Mia” island so not exactly undiscovered but quieter than Skiathos.
Paxos is also gorgeous, there is nightlife but quite “upmarket” (attracts a boaty crowd), you need to get a boat from Corfu.
Kefalonia is lovely and you can fly direct. Some beaches will be busy but it has a lot of beaches and is quite a large island so never too bad. You need to try a few beaches till you find your favourite. It doesn’t have much in the way of traditional Greek towns as an earthquake did for a lot of the buildings (just one town survived). But the modern development is low level and spread out.
Paros and Naxos are nice islands, both a short boat ride from Mykonos or Santorini (these both have airports but are very ££ and busy). Have the classic Cycladic white cube buildings and historical sites to see. May be fairly busy though. Milos is a longer boat ride but much less discovered.
Amorgos is fairly quiet and traditional, hilly, bit tricky to get to (we took an overnight boat from Athens). Same for Folegandros.
Crete is huge, has loads to explore and plenty of quiet areas away from the developed bits. Can be very hot but you can stay in the hills which are cooler and then drive to beaches - your teen may be unimpressed with this though!
What is your DD looking for?