There may not be much that you'll want to buy in Banda Aceh, but the town's two main markets can still make interesting places to wander around.
The entrance to the Aceh Bazaar
The main market, Pasar Atjeh (Aceh Bazaar), is right behind the Baturrahman Mosque. It runs for several square blocks and includes a few newer covered spaces as well as the open air stalls lining the streets. At first glance, most of what is on offer is ladies clothing, but if you get deeper into the market you'll find other things for sale and even a block lined mostly with fresh produce.
A stall selling vegetables in the fresh market
The city's main fresh market is across the river. The center of the market is along Jalan Kantini south of Jalan Supratman. In this market you'll find fresh fruits and vegetables displayed in piles on the pavement. The stall owners are very friendly, and happy to see a western visitor, although their English doesn't extend far beyond “Hey mister! Foto!”
Souvenirs
Skull caps embroidered in the local Acehnese fashion
Banda Aceh gets so few foreign visitors that there isn't really much of a souvenir shopping scene. There are some souvenir shops, especially along Jalan Sri Ratu Syaflatuddin, but these cater mostly to domestic tourists. The main local handicraft seems to be a style of embroidery. The shops are full of it, on caps men wear to the mosque, pocketbooks, wallets and handbags.
The other local product you may be interested in is coffee. Aceh is one of the prime growing areas for coffee on Sumatra, and while it doesn't have a big international reputation, it is highly prized among Indonesians. You will find fancy boxes of coffee in most of the souvenir shops, but you can also find what is probably better coffee for a lot cheaper in the local supermarkets.