Inheritance: Pay off your mortgage?

I've been asked twice in the past few weeks if a person should pay off their mortgage since they just inherited some money. As is most often the case, the answer is, "It depends".
In the first case, this family hated the thought of paying a mortgage for 30 years. The inheritance was just a little larger than the mortgage balance they had, so they could easily use that money to eliminate mortgage payments forever. When we reviewed the financial priorities for this family, we looked at the mortgage, financial protection (insurance), education for their children, and retirement. In each case, the family agreed that the other priorities they had were more important, on paper, than the peace of not having a mortgage. It boils down, as it often does, to the emotional pleasure of not having a mortgage, versus the logic of taking care of more important priorities first, and then paying off the mortgage. For this family, they decided to use some of the money to pay the mortgage balance down while refinancing to a shorter term mortgage. They are putting the rest of the money into retirement and college savings, so they are making progress on three financial goals with this inheritance, rather than putting it all into the house.
Have you been faced with this decision, or if you had this opportunity, what would you do?