3:20 — 20Adam named his wife Eve, because she would become the mother of all the living.
[T - OL ]
The man had already called her 'isshah when she was first presented to him in 2:23. 'issah, however, is not a proper name, only a generic noun recognizing that she is one of his kind. Now he gives her a personal name, Eve, Chawah. The reason given for the name is "because she would become the mother of all the living" (NIV). This places the reason on the lips of Adam as he looks forward to the future in faith that God would still be merciful. This is not inconsistent with the depiction of Yahweh in Gen 1-3.
The verb of existence, hayetah "is/become" in Hebrew is, however, a qal perfect, suggesting a past tense is a more appropriate translation: "because she was the mother of all living" (KJV, NASB,RSV, NRS, NKJ). Taking this reading would suggest that the reason for so naming her derives not from Adam but from the narrator; i.e., it was the author who was explaining why she was so named. Looking back through the history of his Hebrew ancestors, the author certainly had reasons to see why it is appropriate.
You may wish to read the following commentaries-expositions:
Low Chai Hok
©Alberith, 2016