The Hebrew is pretty clear. "Eve" means "life" or "life-source" in this context (it is a substantive). The LXX even uses Ζωή to translate the word. This name (Eve, Ζωή, Life) was given by Adam to his wife, “because,” as the text explains “she became the mother of all living,” that is, because life of all of Adam's descendants was guaranteed to come through the woman.
I do not disagree.
However, I was wondering more specifically if such "life" was an indication of appointment that the redeemer would eventually give come through Eve, rather than all human life. For it is true that life is throughout creation in all living things, and therefore Eve was not the life giver of all living, but specific as the beginning heritage of who would come as redeemer.
But, that be as it may, the other question involved the use of past tense in NASB and KJV in contrast to future tense in the NIV and NLT.