Genesis 3:20

And Adam called his wife's name Eve
Whom he had before named "Ishah", a woman, because taken from him the man, ( Genesis 2:23 ) and now gives her a new name upon this scene of things, which had taken place; which is derived not from "Chavah", to "show forth", to "declare"; as if she was called so, because of her discourse with the serpent, being loquacious and talkative, and telling everything she knew, according to some Jewish writers ^F7; but from "Chayah, to live", as the reason given in the text shows. She is called Aeon "(Aevum)" by Philo Byblius, the interpreter of Sanchoniatho ^F8. The word "Eve" is retained in many Heathen writers, and used to be frequently repeated in the Bacchanalian rites, when the idolaters appeared with serpents platted on their heads ^F9; which plainly refers to the affair between the serpent and Eve; hence Bacchus is sometimes called Evius ^F11: the reason of Adam's giving her this name follows,

because she was the mother of all living;
which reason is either given by Moses, when from her had sprung a numerous offspring, and would be continued to the end of the world; or if given by Adam was prophetic of what she would be; and so the Vulgate Latin version renders it, "because she would be the mother of all living"; and the ground of this faith and persuasion of his, that he and his wife should not die immediately for the offence they had committed, but should live and propagate their species, as well as be partakers of spiritual and eternal life, was the hint that had been just given, that there would be a seed spring from them; not only a numerous offspring, but a particular eminent person that should be the ruin of the devil and his kingdom, and the Saviour of them; and so Eve would be not, only the mother of all men living in succeeding generations, but particularly, or however one descending from her, would be the mother of him that should bring life and immortality to light, or be the author of all life, natural, spiritual, and eternal; and who is called (zwh) , "the life", which is the same word by which the Greek version renders Eve in the preceding clause. It was with pleasure, no doubt, that Adam gave her this name; and it appears that this affair of her being seduced by the serpent, and of drawing him into the transgression, did not alienate his affection from her; and the rather he must needs cleave unto her, and not forsake her, since her seed was to break the serpent's head, and procure life and salvation for them; and by means of her there would be a race of living men produced, which would propagate his species to the end of time: for all living can only respect them, and not other animals, though in some sense they may be included, as our English poet ^F12 hints.