I believe that most airlines offer some sort of bereavement fare, although it may not be the huge discount you'd expect, I've heard that the fares are much more flexible, and do not have the same restrictions as other fares (e.g. weekend stay, etc.)
I've also heard that many airlines offer compassion fares, which you can use to go and visit a terminally ill immediate family member if you wish to spend time with them in their final days.
Both types of fare will probably require some kind of proof of the circumstances, either a doctor's note/phone number to call and verify the illness, or a death certificate copy to verify that someone has passed.