The obvious answer is the mind-body problem, but since you asked I assume you don't consider that non-trivial. So I would add the problems of intentionality and qualia and the now-debunked issues associated with phrenology.

Also, it's important to note the difference between an "example of a non-trivial question which philosophy of mind addresses" and a problem in philosophy of mind which has been (mostly) solved. It could be easily argued that computationalism solves all of the problems above, at least as far as a pretty good number of philosophers are concerned, but these are still issues that are "addressed" in the sense that there is still philosophical literature being written about them.