Mr. Adams is the first person to draw in the lottery, which makes sense alphabetically – but we don't think we're stretching too far to say that it's also because he's the First Man. Yes, "The Lottery" is full of tiny little references to Christian tradition, which we'd be careless to overlook (check out the "Character Analysis" for the Delacroix Family and "The Stool" in "Symbols, Imagery, Allegory"). Remember that Adam (and Mrs. Adam, i.e., Eve) brings about the fall of mankind in the Biblical Genesis story.
Adam and Eve represent a bridge between that early state of grace and the later intrusion of knowledge. In this story, you can read the "early state of grace" as untrammeled human nature, which emerges in the text as the "liberty" of the boys in summer and the primitivism that Old Man Warner fears. The state of knowledge is seen in this story as civilization, and even Warner's vaunted agriculture (remember, one of the great draws of the Garden of Eden was that food grew in profusion without tending by human hands).
We don't want to get too highfalutin' about this, but it is interesting that it's the Adams family that brings up the entire relinquishment of the lottery. Perhaps this is a passing comment on the mythic role of Adam and Eve in advancing "civilization" – an act that has a somewhat ambiguous meaning in Jackson's work. On the one hand, the lottery is clearly hateful – Tess Hutchinson's family and neighbors come together to kill her with stones – but Jackson doesn't seem to have a very positive conception of human nature in its essential forms. Perhaps more "civilization" would just mean more hypocrisy, further disguising of the propensity of human beings to kill one another. For more on the nature of the lottery, check out "Symbols, Imagery, Allegory."