The name 'Eve' comes from the Hebrew word חַוָּה or 'Chava,' a form of the word for life, which is why she is called 'the mother of all living.' She represents the church, and is called 'mother' because this was the first Church of all, and 'living' because of its faith in the Lord, who is life itself.

(Odkazy: Arcana Coelestia 287-291, Genesis 3:20)

362. This verse describes the doctrine of faith which is called Cain, and how because it separated faith from love, it also separated it from charity, the offspring of love. Wherever a Church exists, heresies arise in it as a consequence of people basing their thinking on one particular article of faith to which they attach supreme importance. For man's thought process is such that when he focuses his whole attention on any one matter he makes that more important than any other, especially so when delusion claims it as a discovery of his own, and when self-love and love of the world inflate his ego. In that case everything seemingly agrees with it and supports it, even to the extent of his being ready to swear to it when in fact it is false. In the same way people called Cain made faith more essential than love; and because in so doing they were leading lives devoid of love, both self-love and delusion resulting from it banded together in them.