...from O.Fr. abeste, from L. asbestos "quicklime" (which "burns" when cold water is poured on it), from Gk. asbestos, lit. "inextinguishable," from a- "not" + sbestos, verbal adj. from sbennynai "to quench," from PIE base *(s)gwes- "to quench, extinguish" (cf. Lith. gestu "to go out," O.C.S. gaso, Hittite kishtari "is being put out") (emphasis added).
Like people, every word has lived its own peculiar and unique life. Riffing on my post below regarding words that have the opposite meaning of their etymology, my friend Andy (who did graduate work in Classics, and hence, actually reads Greek) challenged me to help him understand why the word asbestos, whose etymology literally means 'unquenchable' is used today to mean a substance that cannot burn.
With some Googling, I found this (PDF): "First mention of asbestos appeared in the Greek text On Stones, written by Theophrastus, one of Aristotle’s students. Theophrastus referred to a substance that resembled rotten wood and burned (right) without being harmed when doused with oil."
So, Ol' Theophrastus kept pouring oil onto this stuff, but it never burnt, so he kept pouring, but the stuff was never quenched by oil/fire. Hence, it was unquenchable. That's my story and I'm sticking to it (for now).
Andy did some follow-up of his own and provides the following:
Yes, that's one of the more likely explanations. In my research I came across the use of asbestos as permanent wicks in lamps, but never noted the bit about being unquenchable with oil. That Theophrastos citation really belongs in the dictionary entry below, as it's the only cite that explains the meaning under A. The lexicon below is massively comprehensive (if you couldn't tell) so it's odd they missed Theo.
The other possible explanation is II. or "unslaked lime", as quick lime burns underwater. This was a key component in later "Greek fire", but so far I haven't been able to find any ancient source that cites an unquenchable substance (Greek Fire dates to 500 AD, white phosphorus, which also burns underwater, dates to 1600 AD, and sodium, which explodes on contact with water, dates to 1800 AD).
If I had the time and language skill I used to have I would search my CD of all Greek text up to 600 AD for cites of asbestos and then comb thru them, but that would be a day's worth of work I'm pleased that we got close to the meaning in online research and I'm not sure that looking up every instance of asbestos would change anything.
Andy also provided the following reference
A. unquenchable, inextinguishable, “φλόξ” Il. l. c.; not quenched, “πῦρ ἄ.” D.H.3.67, Plu.Num.9; “κλέος” Od.4.584; “γέλως” Il.1.599; “βοή” 11.50; “ἐργμάτων ἀκτὶς καλῶν ἄ. αἰεί” Pi.I.4(3).42; ἄ. πόρος ὠκεανοῦ ocean's ceaseless flow, A.Pr.532 (lyr.); πῦρ, of hell, Ev.Marc.9.43.
II. as Subst., ἄσβεστος (sc. τίτανος), h(, unslaked lime, Dsc.5.115, Plu.Sert.17, Eum.16; “ἄ. κονία” Lyc. ap. Orib.8.25.16.
Henry George Liddell. Robert Scott. A Greek-English Lexicon. revised and augmented throughout by. Sir Henry Stuart Jones. with the assistance of. Roderick McKenzie. Oxford. Clarendon Press. 1940.