As the above poster began, the Eve in EVE Online isn't an acronym; it's a biblical reference. Specifically, it refers to the Eve wormhole, which in the Eve Online background story was the link that brought humanity to the extra-galactic star cluster in which the game takes place.

At the time, the Milky Way had become overcrowded, and its resources were running terribly thin - humanity was on the brink of tearing itself apart. Thus, the discovery of the naturally occurring wormhole to a virgin, resource-rich galaxy was almost seen as divine providence, inevitably resulting in the 'new world' being referred to as New Eden. Naming the wormhole itself Eve was simply fitting (I'm still waiting for them to use Adam for something).

That said, Eve was also, in the story of Genesis, the one who tempted Adam away from eternal life and the eternal happiness of the Garden of Eden, and in some ways the same might be said of the Eve Wormhole - which 'betrayed' humanity by promptly shutting behind them, killing off all but a few of the immature colonies and plunging the inhabitants of the New Eden cluster in a new dark age, while perhaps also throwing the inhabitants in the Milky Way back into their resource wars.