Island crafts and rum cakes!
When I visited Nassau, there are a few shops in the Festival Place (a tourist building) that is right beside where the cruise ships park. In there, you'll find a few shops selling island handicrafts and rum cakes. There were also intricate flower pendants made from fish scales; and they were only $10!
Also, in the tourist area beside the ships, there are plenty of shops for souvenirs. There are also shops for designer replicas as well as real designer goods that are a bit cheaper than US retail prices.