The Fowler Museum on the UCLA campus focuses on global arts and material culture, with strong collections from Africa, Asia, the Pacific, and the Americas. Its changing exhibitions are often more interesting than the phrase "university museum" suggests, especially if you like textiles, ritual objects, ceramics, and the stories behind everyday things.
It pairs well with a Westwood or UCLA visit. The museum is quieter than the big Miracle Mile institutions, which makes it a good choice when you want a serious collection without the crowds.