When did the first woman of the Bible receive her name? I can tell you that when she is named, it is for theologically rich reasons!
Adam gets his name first, from the Hebrew word for man (adam), which is also the generic term used for mankind (both male & female, as in Genesis 1:26-27). So, in Gen. 2:20, and 3:17 God calls him by the proper name, Adam. But up to that point the woman was simply called “female” (1:27), a “helper” (2:18), a “woman” (2:22-23), and a “wife” (2:24-25).
It is only after the Fall into sin, and the pronouncing of the curse by God, that Adam (not God) calls his wife “Eve” in Genesis 3:20 — The man called his wife’s name Eve, because she was the mother of all living. The name Eve is taken from (and sounds like) the Hebrew word for life-giver and resembles the word for “living.”
Why is this theologically significant? It is a bold act of faith on the part of fallen Adam. God had said something in which Adam put his trust and hope, and expressed in naming his wife Eve.
In pronouncing the punishment on the serpent (after the disobedience of Adam and Eve), the Lord spoke words we call the proto-evangel — a forerunner of the Gospel of grace in Christ. In Genesis 3:15 God said to the serpent, I will put enmity between you and the woman, and between your offspring and her offspring; he shall bruise your head, and you shall bruise his heel. Notice the good news here: an offspring (lit. “seed” meaning child) of the woman would conquer the evil one, and be the undoing of the curse of sin!
Believing this to be true, Adam gave his wife the name Eve BEFORE she had conceived or bore a single child! His faith and hope in this gracious promise of God was that certain. When their first child was born, he was named Cain, which might (in modern terms) be translated “Gotcha!” They had the first “seed” who might just be the one to conquer and redeem them from the curse. And after Cain murders his brother Able, and they are tempted to loose hope in the promise, Genesis 4 [below] speaks of Adam and Eve trying and having another son, Seth, and the revival of hope that came with his birth and the birth of his son.
25 And Adam knew his wife again, and she bore a son and called his name Seth, for she said, “God has appointed for me another offspring instead of Abel, for Cain killed him.” 26 To Seth also a son was born, and he called his name Enosh. At that time people began to call upon the name of the Lord.
Eve, the mother of all living physically, but also the name which spoke of spiritual life and that blessed hope of One who would be the promised Seed, our Savior — whom we know as the Lord Jesus Christ, born of the virgin Mary. Praise God for His promise of grace fulfilled.