Thank you for the biblical question, “Who is described in the Old Testament as the mother of all living (Gen. 3:20)?
Genesis 3 says this regarding Adam and Eve “16 To the woman he said, “I will make your pains in childbearing very severe; with painful labor you will give birth to children. Your desire will be for your husband,
and he will rule over you.” 17 To Adam he said, “Because you listened to your wife and ate fruit from the tree about which I commanded you, ‘You must not eat from it,’ “Cursed is the ground because of you; through painful toil you will eat food from it all the days of your life…” 20 Adam[c] named his wife Eve,[d] because she would become the mother of all the living” (NIV).
Adam gave his wife her name, Eve. The reason is that she would become the mother of all living. The context is that she will be the mother of all living human beings. It is through her that the human descendants will come be born as she was the only created woman by God for Adam.
SUMMARY: Eve was given her name by Adam since she would be the mother of all human beings just as Adam would be the father of all human beings.
For more perspectives:
https://www.quora.com/Who-is-described-in-the-Old-Testament-as-the-mother-of-all-living-Genesis-3-20