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)

641. Since this member of the Church needed to be reformed as to that part of the man which is called the understanding before he could be reformed as to the other part referred to as the will, it is described here how those things belonging to the will were separated from those belonging to the understanding, and how the will was so to speak protected and held back to prevent anything coming in contact with it. For if things belonging to the will, that is, to evil desires, had been aroused, he would have perished, as in the Lord's Divine mercy will be made clear later on. With man nothing could be more distinct and separate one from the other than those two parts, the will and the understanding. This I have been given to know plainly, especially from the fact that among spirits and angels things of the understanding flow into the left side of the head or brain, while things of the will flow into the right. The same applies to the face. When angelic spirits flow in they do so as softly as gentlest breezes. But when evil spirits do so it is like a deluge entering with dreadful delusions and persuasions into the left side of the brain, and with desires into the right. The influx of them is so to speak a deluge of delusions and desires.