(Plato, Epigram 28)
“Amethyst is the sober stone, but I am Dionysus the drinker; either it will persuade me not to drink, or it must learn to be drunk!”
I looked into this (just a little bit) after reading this epigram, and found that there was indeed something of a tradition of etymologizing the stone’s name a-methustos (not drunken) in relation to its reputation as a palliative for excessive indulgence:
Nonnos, Dionysiaca 12.380-81
Rhea gave Dionysus the wine-drinker alone the amethyst, that defender against compulsive madness.
μούνῳ δ’ οἰνοποτῆρι Διωνύσῳ πόρε ῾Ρείη
λυσσαλέης ἀμέθυστον ἀλεξήτειραν ἀνάγκης.
Plutarch, Convivial Questions 647b:
“Some even think that herbal amethyst and the stone named after it are so called from their ability to aid against drunkenness, but they err; both are so named from their color.
οἱ δὲ καὶ τὴν ἀμέθυστον οἰόμενοι τῷ πρὸς τὰς οἰνώσεις βοηθεῖν αὐτήν τε καὶ τὴν ἐπώνυμον αὐτῆς λίθον οὕτω κεκλῆσθαι διαμαρτάνουσιν· κέκληται γὰρ ἀπὸ τῆς χρόας ἑκατέρα·
I also found that this etymology was incorporated into the Wikipedia page for the stone, despite the fact that (of the sources that I looked through, at any rate) Plutarch seems to be most specific in his discussion, and flatly denies the etymology. Sed hae sunt nugae.