Dan Brown in his book "The Lost Symbol" describes that soul has mass. The scientist who undertook this experiment in the book measured the mass of a terminally diseased volunteer before and at the exact moment of his death. She found out that he was lighter by an extremely little amount of milligrams after he died so she concluded that soul has a mass.
Another idea could be the scenario below:
In Newtonian physics, when an object is moving with a certain speed, its mass remains constant, or the difference in mass is negligible.
However, according to Einstein, as the speed of an object increases, so does it's mass. Consequently, when the object reaches speed of light it's mass equals to infinite.
When a person is alive, his body consists of 'objects' that are in motion: blood cells, nerve impulses, the heart etc.
But when he dies most of these 'objects' stop moving.
Maybe that is why there is such a difference in mass before and at the very moment of death!
This is a simple idea and may not correspond to reality.