Everyone knows the story: a bite of forbidden fruit lodged in Adam's throat, and men have carried the lump ever since. It is an old story, but it explains the name rather than causing it.
Wikipedia quotes the 1662 English translation of a 1651 anatomy by Thomas Bartholin. "The common people have a belief," it says, that a piece of "that fatal Apple" stuck in Adam's throat. Merriam-Webster cites a 1707 treatise by John Purcell with a fuller version. God left the bump in all Adam's male descendants as a reminder, and it showed less in women because Eve's offense was thought smaller. Wikipedia notes that the story appears nowhere in the Bible or other Jewish, Christian or Islamic writings.
The name came first. Merriam-Webster explains that medieval writers used Latin pomum Adami for several fruits, among them the pomegranate. Etymonline adds an early 15th-century English use for a kind of lime, whose dents were thought to look like Adam's tooth marks.
How the fruit name reached the throat is less certain. Merriam-Webster cites the etymological dictionary of Elmar Seebold. It reports that medieval Arab medical writers called the bump the "pomegranate." Europeans may then have adopted pomum Adami as its Latin name. Caspar Bauhin's anatomy of 1590 lists both "pomegranate" and "Adam's apple" for this part of the larynx. Merriam-Webster also finds pomum Adami in Gabriele Falloppio in 1584, and Wikipedia in a Paris anatomy text of 1600.
English followed soon after. Merriam-Webster dates "Adam's apple" for the throat to 1625. Etymonline gives a later date, 1731.
A third theory says the Latin mistranslated a Hebrew phrase. Wikipedia credits the linguist Alexander Gode with proposing it in 1968, though Seebold, cited by Merriam-Webster, dates the idea to the 19th century. The claim is that tappuach ha adam meant "the man's bump," since Hebrew adam means "man." Etymonline still lists it as possible. Merriam-Webster calls it "specious," because no such phrase has been found in pre-modern Hebrew.
The medical name is plainer. Wikipedia says prominentia laryngea, the laryngeal prominence, was introduced in 1895.