I was interested in the story about Sarah Palin, a journalism major in college, who couldn't name a newspaper or magazine that she ever reads. She generalized that she likes "all of them," but asked to be more specific, chose not to hurt anyone's feelings by naming particular ones and possibly leaving someone out.
Really, though, if it seems like a test to prove that you have at least the intelligence of a moron, it seems like you would risk hurting someone's feelings by naming a few of the more well-known publications. You know, you might say, Time or U.S. News and World Report -- everyone's heard of those. Maybe Wasilla has a local paper, a weekly shopper, a school newsletter, something. It seems, you're the governor of the state, the capital city probably has a major daily, which perhaps you read once in a while.
Then there are the rather obvious newspapers across the country. One notable one, from a swing state no less, would have to be the Dacron Republican-Democrat of Ohio. Just say you read it -- you prove you're well-informed, you make the folks of Ohio happy, they vote for you, you're president! (Statistically speaking, that is, she would likely be president sometime in the next eight years.)