The "First Crossing" is a term that refers to the 15-year, one-way journey that the original Pern colonists made from Earth to Pern. Most of the colonists were in suspended animation for the entire 15-year trip. The crew of the colony ships were awake for 5-year shifts, and asleep the other 10.
The dolphins call this the Great Sleep. The Name Song remembers the names of the 25 dolphins who made this crossing aboard the Bahrain.