There are two main or obvious possible reasons:
- Achilles died of a wound to the heel, from a poisoned arrow shot by Paris/Alexander. This is sometimes fabled to be the only spot where he could be wounded, thanks to a magical invulnerability elsewhere due to immersion in the Styx (which failed to wet his heel because his goddess mother was holding him by that part).
- Achilles pierced the slain body of Hector between this tendon and the underlying bones (tibia and fibula) so as to pass through the thongs by which he dragged the body behind his chariot.
The first would seem to be the more popular notion, but it is completely un-Homeric—the invulnerability thing in particular would make nonsense of Iliad 18, and neither epic mentions the wound in the heel at all—while the second is authentically Homeric (Iliad 22.395–404).
Brewer’s 1898 (courtesy of Bartleby) is no help, nor is Free Dictionary, and Wikipedia hedges its bets (and ignores the question in Talk). OED cites the following, which clearly favors #1, but nowhere casts its authority behind that origin-story:
1703 tr. P. Dionis Anat. Humane Bodies Improv’d ii. x. 422 That String is call’d the Tendon of Achilles, because, ’tis said, he dy’d of a Wound in that Part.
P.S.: Irene, commenting on a question regarding whether the expression needs an apostrophe, alleged that the Greek origin involved the adjective Ἀχίλλειος, but Liddell & Scott indicate that the tendon in question was simply called ὁ τένων ὁ ὀπίσθιος, the rear tendon, while their entry for Ἀχίλλειος does not mention any tendon. Wikipedia traces the expression to a decade before the OED example previously given:
The oldest-known written record of the tendon being named for Achilles is in 1693 by the Flemish/Dutch anatomist Philip Verheyen. In his widely used text Corporis Humani Anatomia, Chapter XV, page 328, he described the tendon's location and said that it was commonly called “the cord of Achilles”, now also called “tendo achillies” by anatomists. (“quae vulgo dicitur chorda Achillis”)