Two of the most common screen presidents have been Richard M. Nixon and Abraham Lincoln, both wellsprings of human and political drama you couldn't really fabricate.
The new Steven Spielberg-directed "Lincoln," starring Daniel Day-Lewis, works (and works well) because it's not trying to give us the whole story, one president's rise and fall. It's confined to the final four months of his life, as the Civil War was ending and the 13th Amendment to abolish slavery was being debated. And while Spielberg and screenwriter Tony Kushner ensure plenty of iconic scenes (including Lee's surrender to Grant at Appomattox), one of the reasons the film works, I think, is because it's less interested in Great Moments of History than in the grubby workaday process of politicking.
In any screen portrait of a president, you don't want what you think you already know.
You want to see the human being behind the façade (or the penny, or the fiver). You want to see the contradictions at work in any great statesman. You want engagement – the filmmakers', and your own.