Adam called the name of his wife Hava, for she was the mother of all the living. (Breishit 3:20)
Two observations on this.
1) One instinctively connects the name Hava (חוה) to the last word of the verse (חי). The problem with this is clear: Hava is not all living beings, she is simply the MOTHER of all living beings (well, the human ones). Shouldn't her name suggest that she's the mother, rather than implying that she's simply one of those living things herself?
For this reason, I think her name must come from a different source. It is a derivation not of חי, but of חיה. Haya is a similar-looking word with a different meaning (though clearly, they are related). As explained by the Rashbam on Breishit 18:10 (any guesses who I'm doing for shnaim mikra this year?) חיה refers to a woman who has recently given birth, similar to יולדת.
If so, then the verse is much more easily understood: “Adam called the name of his wife 'Birthing Mother', for she was the mother of all the living.”
2) Why then is her name Hava, not Haya?
The name Hava sounds a lot like a more familiar name – that of God, YKVK. By examining that name, we can understand Hava's as well.
The meaning of YKVK is best explained through another name of God – “Ehyeh Asher Ehyeh” (Shemot 3:14). I once heard a suggestion that this name is of the form “Bond, James Bond” and it proves that God's first name is Asher! But more seriously, Ibn Ezra (3:14) explains that “Ehyeh” is God's name (thus the introduction “Ehyeh sent me to you” that appears later in the verse), and “Asher Ehyeh” is an attached explanation (“Who is always existing/present*”). Of course, the Hebrew word Ehyeh is in the first person – appropriate for when God is talking about himself. When humans talk about God in the third person, the appropriate form would be Yihyeh. This is almost the same as the well-known name YKVK, and Rashbam and Hizkuni (3:15) argues that this is how the name YKVK was formed.
The only difference between Yihyeh and YKVK is that the yud in the middle is replaced with a vav. Thus the word no longer takes a grammatically correct form, and it is clear that it represents a unique name, rather than just being a verb. I suspect that God's name is not the only name with this yud-to-vav transformation. It seems to have also occurred in the formation of Hava's name, transforming it from the dictionary word חיה to the name חוה.
* For a full discussion of the verb "lihyot", its primary meaning of "appearance" rather than "being", and the implications regarding God's name and role in Tanach, see R' Yoel Bin Nun, "Active and existential being in the Bible: a linguistic interpretation of the name Hashem", Megadim 5:7-23. As an aside, I once heard or read R' Yaakov Medan discuss the implications of this meaning for our relationship with God: it is encouraging that we are guaranteed Divine presence in the world, but also more threatening, in the event that we sin.