If you love 'experiential learning', the Galata museum is a great winter option for exploring the maritime history of the port of Genoa. With great 'hands-on' exhibits, and a beautiful roof top observation deck, this museum is one of the best in Genoa. And at only 11 euro, it is priced quite effectively.
Make sure you don't miss the "Tempest" exhibit -- a 4D experience of surviving a massive storm in a tiny life boat off Cape Horn. You can also pay a supplemental fee to explore the submarine parked in the harbor in front of the museum - but if you suffer from claustrophobia at all, I'd steer clear, as the space inside is incredibly small.