One of the many fascinating exhibits in Coimbra's National Museum
Museu Nacional Machado de Castro in Coimbra is very close to the historical university and well worth a visit. Recently remodelled, it sits on top of a 2000-year-old Roman cryptoporticus (great for escaping the summer heat).
Over the next three floors, you can explore the history of Coimbra and Portugal through stone sculptures, gold jewellery, decorative ceramics, ancient art, exotic furniture and religious reliquaries.
I've been twice since it reopened in 2012 and still haven't seen everything.
Even if you're not really interested in museums, it's worth going through the courtyard to the café terrace to great, free views over the cathedral and the narrow streets spreading downhill to the river. They do a cheap lunchtime menu (about 7 euros), too.