The name Eve in the Bible
The name Eve shows up too in the Greek New Testament, where it is spelled Ευα; Eua (2 Corinthians 11:3, 1 Timothy 2:13).
Etymology of the name Eve
Most (if not all) translators and interpreters derive the name Eve from the verb חיה (haya), meaning to live or have life:
In Genesis 3:20 it reads that Eve was named this way because she was אם כל־חי; mother of all life. The Hebrew word אם ('em), mother, comes from the same root as אמה ('amma), mother city, cubit, tribe/ people; hence the phrases Mother Babylon and Mother Jerusalem. The phrase "all life" returns six times in Scriptures and never just mankind is meant (show me). Hence the "mother of all life" is the biosphere; all living things.
And hence the NOBSE Study Bible Name List, Jones' Dictionary of Old Testament Proper Names, and even BDB Theological Dictionary report in dubious unison that the name Eve means Life.
But note the difference between the verb that means to live and the name Eve: חיה and חוה. The verb is spelled with a yod but the name is spelled with a waw. These two letter sometimes interchange, but this verb, and all its many derivations are always spelled with a yod. So yes, Eve is the mother of all life, but her name simply doesn't mean life. It's a leap of critical neglect to assume that when parents call their baby Bob, and say, "because that was the name of our dog," they are doing this because they mean to propone that the name Bob means dog.
The name חוה (Eve) is identical to the Hebrew root group חוה (hwh I, II & III):
Eve does not mean 'the girl of the two,' and is also not the model for all wives bound to goof up (Keil and Delitzsch' Commentary on the Old Testament). Man and woman are both under the label Adam, and their drive to group up lodge in Eve.
The name Eve denotes the collectivity that is common to the behavior of living things. We should never forget that multicellular creatures, even humans, are in fact highly efficient colonies of single cellular creatures.
The name Eve means Gathering, Declaration or Humility, because she was the mother of all of life.