Such a person could RUN from anywhere in the US. Being successful and getting ELECTED, though, is an entirely separate matter. Offhand, I'd say, the best bet would be in the northeast, New York or somewhere in New England, very likely. The Left Coast (California - Oregon - Washington) is also a possibility, as the electorates there are sufficiently liberal-minded to consider a non-theist and just maybe put him or her in office.
The Deep South for the large portion is OUT, for reasons which should be intuitively obvious, though North Carolina does have at least one state official who is an avowed atheist (and who also had a recall effort aimed at him once the news got out ... by guess who?!?). I'd love to think that my own home state of Ohio could support such a candidacy, but as red (and sometime redneck) as things can get in the rural areas of the Buckeye ... well, someone could run, but how far he would get would be debatable.
Pete Stark (D), US House of Representatives, California, 1973-2013.