I visited Hamburg 2 years ago in October - and it rained almost every day. The tour bus had all windows fogged up. They gave us papertowels to wipe the windows. Didn't help. If you visit between October and February, this could be your experience. Here are a few rainy day suggestions:
Visit Miniature World in the Hafen City. Incredible what they have created in miniature. Even an airport with planes taking off. Be prepared for long line-ups. Everyone else is heading there.
Visit the Old Elbetunnel at St Pauli Landungsbruecken. It's free and it gave me goosebumps to think you are walking under the River Elbe.
On Sunday morning, visit the Altona Fishmarket (all indoors). Oct 5:00 am to 9:30 am, Nov to Feb 7:00 to 9:30 am. Vendors holler out their wares and prices. Great seafood meals and deals on about anything.
Starting in Feb 2017, tours are offered of the new Elbphilharmonie Concert Hall. Individual tickets are sold for tours in German only. Group tours can be arranged in other languages as well.
Are there any other sites to visit - any other recommendations?.