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. Thus Adam bears the name of the dying body, Eve, of the living soul. Though for her sin she was justly sentenced to a present death, yet, by God's infinite mercy, and by virtue of the promised seed, she was both continued in life herself, and made the mother of all living. Adam had before called her Isha, woman, as a wife; here he calls her Evah, life, as a mother. Now, 1st, If this name were given her 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, and an assurance to them, that, notwithstanding their sin, 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. 2d, If Adam did it of himself, it was an instance of his faith in the word of God.
Related Commentaries of Genesis 3:20
Genesis 3:20
20 And Adam called his wife's name Eve;e because she was the mother of all living.