The most worthwhile thing to bring home from Naxos is food. Graviera cheese — the island's excellent aged cheese, nutty and firm, made from sheep's and goat's milk — travels well and costs roughly half what you'd pay for the same quality in Athens. Vacuum-packed graviera and the sharper arseniko keep for weeks. Several shops on the main street of Hora and inside the market specialise in local produce: dried herbs, honey, local olive oil, and the preserved citrus products made from the citron tree.
Kitron liqueur is the obvious bottle to pack. The Vallindras distillery in Halki has been making it for generations, and you can buy directly from them after a free tasting. It comes in three varieties — green, yellow, and clear — distinguished by sweetness and strength. The green, made from young citron leaves and unrefined, is the most distinctively flavoured. Most Hora shops also stock it, but the distillery price is better and the visit is worth making anyway.
The Kastro and the lanes below it have the usual archipelago of jewellery shops, ceramic studios, and galleries. Quality varies. A few potters and silversmiths in the old town do work that's actually rooted in local tradition rather than island-tourism generic; it's worth looking at several places before buying. Naxian marble — the same stone that supplied ancient sculptors across the Aegean — appears in small decorative pieces, bookmarks, and objects at prices ranging from tourist trinket to serious craft work.
For practical shopping, the market behind the main waterfront street has supermarkets, pharmacies, and hardware shops that serve the local population rather than visitors. Prices there are accordingly reasonable.