Genesis 3:20

As justice and mercy were combined in the divine sentence; justice in the fact that God cursed the tempter alone, and only punished the tempted with labour and mortality, mercy in the promise of eventual triumph over the serpent: so God also displayed His mercy to the fallen, before carrying the sentence into effect. It was through the power of divine grace that Adam believed the promise with regard to the woman's seed, and manifested his faith in the name which he gave to his wife. חוּה Eve, an old form of חיּה, signifying life (ζωή, lxx), or life-spring, is a substantive, and not a feminine adjective meaning "the living one," nor an abbreviated form of מחוּה, from חוּה = חיּה (Gen 19:32, Gen 19:34), the life-receiving one. This name was given by Adam to his wife, "because," as the writer explains with the historical fulfilment before his mind, "she became the mother of all living," i.e., because the continuance and life of his race were guaranteed to the man through the woman. God also displayed His mercy by clothing the two with coats of skin, i.e., the skins of beasts. The words, "God made coats," are not to be interpreted with such bare literality, as that God sewed the coats with His own fingers; they merely affirm "that man's first clothing was the work of God, who gave the necessary directions and ability" (Delitzsch). By this clothing, God imparted to the feeling of shame the visible sign of an awakened conscience, and to the consequent necessity for a covering to the bodily nakedness, the higher work of a suitable discipline for the sinner. By selecting the skins of beasts for the clothing of the first men, and therefore causing the death or slaughter of beasts for that purpose, He showed them how they might use the sovereignty they possessed over the animals…

God having named the man, and called him Adam, which signifies red earth, Adam, in further token of dominion, named the woman, and called her Eve, that is, life. Adam bears the name of the dying body, Eve that of the living soul. The reason of the name is here given (some think, by Moses the historian, others, by Adam himself): Because she was (that is, was to be) the mother of all living. He had before called her Ishah - woman, as a wife; here he calls her Evah - life, as a mother. Now, 1. If this was done by divine direction, it was an instance of God's favour, and, like the new naming of Abraham and Sarah, it was a seal of the covenant, an assurance to them that, notwithstanding their sin and his displeasure against them for it, he had not reversed that blessing wherewith he had blessed them: Be fruitful and multiply. It was likewise a confirmation of the promise now made, that the seed of the woman, of this woman, should break the serpent's head. 2. If Adam did it of himself, it was an instance of his faith in the word of God. Doubtless it was not done, as some have suspected, in contempt or defiance of the curse, but rather in a humble confidence and dependence upon the blessing. (1.) The blessing of a reprieve, admiring the patience of God, that he should spare such sinners to be the parents of all living, and that he did not immediately shut up those fountains of the human life and nature, because they could send forth no other than polluted, poisoned, streams. (2.) The blessing of a Redeemer, and promised seed, to whom Adam had an eye, in calling his wife Eve - life; for he should be the life of all the living, and in him all the families of the earth should be blessed, in hope of which he thus triumphs.

3:20-24 Soon after they were judged for their sin, Adam and Eve were banished from the garden. 3:20 Eve (Hebrew khawah) sounds like a Hebrew term (khayah) that means “to give life.” Following God’s pronouncement of Adam’s impending death (3:19), Adam expressed hope by giving Eve a name associated with life. Adam’s naming of Eve in such close proximity to 3:16 may suggest that the narrator views it as Adam’s first act of ruling over the woman after the Fall (see study note on 2:19-20).