Two things to bring home from Zanzibar: spices and kanga.
For spices, the best buys are cardamom, cloves, vanilla pods and the dark red pilau masala mix — Darajani Market sells them at local prices in the morning. The little shops along Gizenga Street and Hurumzi sell pre-packed gift assortments at tourist prices but with prettier wrapping; the contents are the same.
Kanga are the printed cotton wraps that Zanzibari women wear, sold in matched pairs with a Swahili proverb running along the edge. Pick a saying you like — the better shops will translate them for you. The Memories of Zanzibar shop on Kenyatta Road and Cap'n's House at the harbour end of Shangani Street are reliable for both kanga and the related kitenge prints.
Avoid anything described as antique unless you are sure: most "Omani chests" sold to tourists are recent. Skip the carved-stone Maasai figures, which are Tanzanian mainland tourist tat, and look instead for the local Tinga Tinga paintings, which are colourful, cheerful, and at least from this part of the world.