A person dying is a person getting closer to their god(s) or a lack thereof. It is at these times that the portents of Destiny can reveal itself to that person to present to the world around him, or her. That or maybe a person should be allowed to get the last word in a discussion every once in a while.
A Character making a point at listening to another Character's dying words gets a Fame card. He might be getting a hint as well but that is dependent on the GM/Director.