The narwhal's name probably means "corpse whale," and the sea's unicorn may owe that grim label to the color of its skin.
English borrowed the word in the 1650s from Danish and Norwegian narhval. Etymonline says that form is probably a metathesis, a swap of sounds, of Old Norse nahvalr. The second half is easy: hvalr means "whale." The first half, na, means "corpse." Etymonline connects it to the same root as English need.
Why a corpse? Etymonline suggests the animal's whitish color, which resembles a dead body. Wikipedia adds the grey, mottled skin and the narwhal's habit of floating motionless at the surface. Both are offered as likely explanations. No early source records what the namers were thinking.
Not everyone accepts the corpse. Etymonline reports that Winfred P. Lehmann, a professor of Germanic linguistics, called it a folk etymology. In his view nahvalr was a West Norse term for a whale marked by a long, narrow projection. That would point to the tusk, not the skin. Etymonline takes this view from Barry Lopez's book Arctic Dreams, so it reaches us at second hand.
The tusk carries the animal's other name. It is a canine tooth that projects from the left side of the upper jaw, and it can grow 1.5 to 3 meters long. Linnaeus described the species in 1758 as Monodon monoceros, Greek for "single-tooth single-horn."
That horn had a long career before science named it. Wikipedia says Vikings and the Greenland Norse likely traded narwhal tusks, which reached markets as far as the Middle East and East Asia. Medieval Europeans believed the tusks were unicorn horns and used them against poison and disease.
The Norse left English one more strange whale name. Etymonline dates walrus to the same decade, the 1650s. It came from Dutch, probably reshaping an older Scandinavian whale word so that it seemed to mean "whale horse."