The name comes indirectly from Greek mythology: After receiving a prophecy that her young son, Achilles, would die in battle, the goddess Thetis dipped him into the magical, protective waters of the river Styx. However, she held him by his heel which was not immersed and thus remained vulnerable. Years later, as luck would have it, Achilles was fatally wounded during the Trojan War: a poison arrow in the back of the foot, a region henceforth known as Achilles heel (and so, metaphorically, an "Achilles heel" is a seemingly insignificant but in fact critical weakness).
It wasn't until 1693 that the term Achilles tendon appeared (chorda Achillis in the original Latin), so named by the resourceful Dutch anatomist Philip Verheyen while dissecting his own amputated leg.
Thetis dipping Achilles in the river Styx.
Philip Verheyen dissecting his Achilles tendon. Artist: anonymous; from the collection of Pieter Deheijde, Amsterdam