In 1693, the Dutch anatomist Philip Verheyden performed a dissection of his own leg (after it was amputated). Upon reaching the tendon we now call the Achilles Tendon, he jokingly (or poetically) referred to it as chorda Achillis, or "Achilles' sinew". This was the first recorded connection between a real part of human anatomy and the myth of Achilles, and several sources credit Verheyden with coining the association. (EDIT: This account appears to be apocryphal; apparently in his writings he credits several colleagues with coming up with the name.)

As for specifically calling it the Achilles tendon, that name was coined (in Latin, as tendo Achillis) in the early eighteenth century by German anatomy professor Lorenz Heister.

As to other mythological body parts:

• The atlas vertebrae holds up the skull, just as the titan Atlas held up the sky in Greek mythology
• The Latin lympha is derived from the Greek nymphe; lympha meaning "clean and pure water" after the Greek story of the nymphs. Our bodies' own lymph is so named.
• See this paper investigating (with good citations) a whole bunch more.