Short answer
Good local buys include Azorean tea from Gorreana or Porto Formoso, pineapple liqueur and preserves, local cheeses, and hand-woven textiles. Ceramics and wicker also make nice keepsakes. Buy from local producers and markets where possible — it supports the island and you get something authentic. Check what you can legally bring home (food and liquids) before you pack.
General information, not legal or tax advice. Rules and amounts change: confirm with official sources.