Genesis 3:20 - "20 Adam named his wife Eve, because she would become the mother of all the living." - NIV

Introduction
Genesis 3:20 calls Eve the "mother of all living." Most interpret this to mean all people, and some include all life. This proves to many Christians that the Earth is young, because Eve could not be the mother of all people unless all people physically descended from her. On the other hand, why would the mother of death be named the mother of all living?
What makes this verse hard to interpret is that the Hebrew is not properly translated in most common English Bibles. Here is a sample of the variety of translations for the verse [emphasis added]...
20 And Adam called his wife’s name Eve,
because she was the mother of all living. - NKJV
20 And the man calleth his wife's name Eve:
for she hath been mother of all living. - YLT
20 Adam named his wife Eve,
because she would become the mother of all the living. - NIV
20 Then the man—Adam—named his wife Eve,
because she would be the mother of all who live. - NLT
The Mother of Humankind?
According to Dr. J. H. Hertz, the editor of "Pentatuech & Haftorahs" by The Soncino Press, the translation mother of all living "... is incorrect. Render, the mother of all humankind. Otherwise, some word must be supplied after 'living', so as to exclude animal life (Onkeos, Saadyah). W. Roberston Smith has shown that the word Chay in the text, is the primitive Semitic word for 'clan'; Eve was the mother of every human clan, the mother of mankind." He adds that 1 Samuel 18:18 uses the word Chay in the same manner, where it is translated families or people.
Who Are the Living?
Mapping the world's development to the literal wording of scripture demonstrates that mankind existed long before Adam and Eve were born. But how does this make sense, when scripture so clearly tells us that Eve is the mother of all humankind? The answer is that it does not say it so clearly.
Look again Dr. Hertz's explanation. Even his interpretation of "humankind", rather than proving that all humans came from Eve, assumes it. Clearly, if Eve were the first woman, then he would be right; if however, God created humans 200,000 years before Eve was born, then not all humans descended from her. What we can conclude from Genesis 3:20 that Eve's fate was to be the mother of all human "clans"; until her Fall and banishment from the Garden, she wasn't.
When Adam Named Eve
Interestingly, Adam's naming of Eve is his only recorded response to God's punishment. Prior to this, Adam was born to people and (archaeologically speaking) grew up learning to till the ground. God then put Adam in the Garden, created Eve from his rib (though she did not have her name yet), and then banished them both from the Garden after Adam and Eve sinned. It was only during this banishment that Adam named Eve, saying that she would be the mother of those-who-live-in-clans... or perhaps more specifically, civilization. In fact, as translations get more "modern", they increasingly add the sense that Adam was looking toward the future; they all seem to agree that this future began upon Adam' and Eve's sin and banishment.
Before Adam named Eve, he was banished, but had not yet left the Garden. He and Eve were still naked, both physically and spiritually. After he named her, God made them clothing from animal skins within the Garden, and then banished them. It was in between these events, naked and exposed, tramatized by his mistake, when Adam realized that humankind -- from whom God had separated Adam -- would see Eve as their mother.
Eve's "Children"
If Eve is the mother of all humankind, does "all" mean "all"? Eve came from Adam's rib; he did not come from her womb. So if motherhood in v. 20 refers to physical descendency, then "all" does not mean "all". But hasn't science proven that we all descend from some common woman in the far past? Yes. Research traces back all of humankind to a single hypothetical women that many call "Mitochondrial Eve". However, she is estimated to have lived around 200,000 years ago, when homo sapiens begin to appear in the fossil record. In other words, Biblical Eve is not the physical mother of us all; Africa, China, India, and elsewhere all descend from Mitochondrial Eve, but not Biblical Eve.
It turns out that Adam and Eve were merely two people out of many throughout the world when she was named. That said, Adam and Eve were leaving the Garden as the first and only people to know God and right from wrong. Knowing how easy it would be succomb to temptation, perhaps that was on Adam's mind as he looked at Eve naked (God had not yet made them clothes). He realized that as knowledge of right and wrong spread, so would sin spread throughout the world... and it did! In this respect, Eve was not the mother of humankind yet, but she would be, just as Adam said.
Furthermore, God's pronouncement of enmity between the serpent's seed and Eve's may have indicated to him what was in store for humankind. We now know Eve's "seed" was Jesus. She was His "mother" in some respect, and Jesus died for the sins of all. So, not only did all sin start with Eve, but our salvation came through her seed... Jesus. This makes Eve the mother of all sinners and of all redeemed -- the mother of the dead (those separated from God by her sin), and now the living (those in communion with God, through her seed, Jesus) -- the mother of all humankind. But notice this: if humankind is made of the dead (separated from God) and the living (reconciled to God through Jesus), Adam only named the living. Could he have meant the mother of all the saved through her Seed, Jesus? Interesting.
Looking to the Future
If the Earth is old and Adam's insight was about temptation, sin, and redemption, then it explains the progression of translations for the verse that we see above. As we move away from an "informed" translation that betrays an assumption of physical descendancy, our progressively updated translations expose the insight that Eve was not the mother of humankind before the Fall, but she "would become" (NIV) or "would be" (NLT) in the future, the mother of all living, through Jesus.
The YLT says this from the perspective of Moses, who had had been given the Law just before he wrote this, and so understood that Eve "hath been" (YLT) the mother of all human clans. "Hath been"? It is an odd perspective if Moses' only point is that Eve had children. Would childbirth not continue after the Law? Of course it would, but by delivering God's Law to the Children of Israel, Moses may have assumed that his people would be saved from their sin, a profound insight into Eve's motherhood over sinners, even if the world was already populated and subdued when she was named (Genesis 1:28).
Conclusion
Seeing the spiritual nature of Eve's motherhood does not prove that the earth is old. Without the physical evidence to back it up, the passage is rather silent on the issue. On the other hand, the spiritual aspect provides the completeness that a straightforward reading of the verses suggests. Given the context in Genesis 3 of temptation (Gen. 3:1), sin (Gen. 3:6), punishment (Gen. 3:14-15, 16, 17-19), and prophecy of redemption (Gen. 3:15), the theological implications for our sinful nature and need for Jesus really come out when the spiritual element is realized. From a Young Earther's perspective, it is but one aspect to consider; for an Old Earther, it is all there is to see.