Tragedy Often Leads to Untrimmed Facial Hair
A beard is usually known to grow naturally and with care; they can grow to be presentable. A "Grief Beard", however, is a beard that has grown rugged and unattended for an amount of time because of a terrible event that causes the owner of said beard to fall into a depression. This depression is so deep that the owner stops caring about little things like shaving and therefore, the "Grief Beard" appears.
However, games generally do not simulate the actual growth of a beard and therefore just magically summon a beard onto the poor chap whose family conveniently died at the hands of cruel game developers.