Try to avoid the typical souvenirs (made in china), although if you want them there are some small stores near the ports of Aigiali and Katapola.
If you ask me I suggest to try some local foods and drinks to bring back home. The local honey, the pasteli dessert are some nice gifts. The local cheeses (kefalotyri and myzithra) are also very tasty, try them even if you cant carry them back (you can eat them on the boat of course! lol)
What I loved in Amorgos was the “psimeni raki” and “rakomelo” both great drinks, you will probably served one after some lunch in a local restaurant. Rakomelo is honey boiled with raki, sugar and clove-gillyflower. Many places and markets sell them. A big bottle of psimeni raki costs about 7 euro.
There is a great market here. You can see it in the pictures. It was called Amorgos market. The fruit was sweet, ripe, and juicy! I would recommend the fruit and yogurt on any of the islands!