Eden is Day's youngest brother and the only other surviving member of his family by the end of the book. Because Eden is all that Day has left, he and June decide that they must go to the waterfront and save him from further experimentation. Eden also represents Patient Zero—he's the first person that the government has infected with a new strain of the plague.
It's worth noting that Eden's name is a biblical reference. The garden that Adam and Eve are in before Eve eats the apple and all knowledge breaks loose is called Eden and, after the little incident with the apple, Eden is the place that Adam and Eve are cast out of. Eden is supposed to be perfect. So perfect, in fact, that the Christian Bible doesn't just open with it—it ends with a promise of its return, too. Given how awful things generally are in Legend, Eden and his virus may be a sort of twisted take on the biblically promised return to Eden… or his name might be a stark reminder of how far humanity has fallen away from perfection.