From Grateful Dead FAQ, Legend says that Gerry Garcia, leader of the band The Grateful Dead, found the phrase in a Funk and Wagnall's Dictionary
The dictionary entry reads along these lines:
GRATEFUL DEAD: The motif of a cycle of folk tales which begin with the hero coming upon a group of people ill-treating or refusing to bury the corpse of a man who had died without paying his debts. He gives his last penny, either to pay the man's debts or to give him a decent burial. Within a few hours he meets with a travelling companion who aids him in some impossible task, gets him a fortune or saves his life. The story ends with the companion disclosing himself as the man whose corpse the hero had befriended.(Funk & Wagnall's Dictionary).
The name has also been attributed to this quote, though it's generally believed that they came across this one later:
"We now return our souls to the creator, as we stand on the edge of eternal darkness. Let our chant fill the void in order that others may know. In the land of the night the ship of the sun is drawn by the grateful dead."