Sin must be atoned. This something mankind cannot do. Adam and Eve tried to cover their sin and shame when they covered themselves with leaves. (Genesis 3:7) However, that’s not how sin must be covered. Atonement requires a reconciliation by blood. (Exodus 29:36; Hebrews 9:22; Ephesians 2:13-14; 1 John 1:7) And that’s what God provided.

And Adam called his wife’s name Eve; because she was the mother of all living. 21 Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them. 22 And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever: 23 Therefore the LORD God sent him forth from the garden of Eden, to till the ground from whence he was taken. 24 So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life. (Genesis 3:20-3:24)

Verse 20 simply appears to be an informative statement conveying that the woman’s name was Eve. The name “Eve” is a Hebrew name which means “life”. As the text states, this is appropriate because all human beings came from her. She was the first mother.