For centuries the center of the Baltic amber trade, Gdansk is still the place to buy it. You'll find no shortage of amber dealers in town. The biggest concentrations are on the main street Dluga and along quieter Mariacka, a couple of streets over. Before buying try to educate yourself a bit about quality amber. While the majority of the dealers are reputable, amber fakes abound so always watch carefully that you're buying the real deal. The Amber Museum is a good place to start to learn about amber. One legitimate gallery with some beautiful pieces is L Galeria, Mariacka 23/24.
Another purely "I got it in Gdansk" gift is Goldwasser vodka. While the sweetish taste is not to everyone's liking, who could pass up flakes of gold in their cocktail? You can buy a gift box at the Goldwasser restaurant.