Either of the Roosevelts, honestly. While they both had downsides (Teddy was jingoistic and racist--he did happily take part in the Spanish American War prior to becoming president; and FDR was pretty authoritarian, what with his court packing plans and putting Japanese-Americans in internment camps and serving four terms), they were still considered some of the greatest presidents the U.S. had. Teddy Roosevelt is probably the most alpha male to have ever existed (and this is coming from someone who thinks the concept of alpha and beta males is stupid; he was rich, strong, fit, skilled, etc.), and is WAY more of a man (and overall respectable human being) than Donald Trump and his base could ever wish to be (no offense to those who actually support Trump here). Hell, Trump could be considered an anti-Theodore Roosevelt, in my opinion. Teddy may have been a Republican, but Republicans were way different back then, back when they weren't the shitstain of a party they are now. He was pretty progressive, being a big conservationist, and regulated corporations (Trust-Busting Teddy). Square Deal, anyone? He did some imperialist stuff, like with the Panama Canal, but despite that, he's still an amazing president.
Moving on to FDR, he helped take the country out of the Great Depression with his New Deal (though some argue it made the Depression worse, and that WWII alone helped pull the U.S. out of the Depression). He's the progressive icon of Democrats as a whole, with his huge programs that were part of the New Deal. The fact that he was able to manage the disasters that were the Great Depression and WWII throughout his presidency, all while being wheelchair-bound due to polio, is pretty impressive. He's definitely the type that would inspire Democrats, especially the progressives.
FDR would be my number 1 choice, with Teddy being number 2.
@LesserAngel Poor Jimmy Carter. He got screwed over real badly by the events of the 70's, what with the oil crisis, and especially the Iranian Revolution. Speaking of the 70's, they were a terrible time politically. There was the end of the Vietnam War, Watergate, Nixon's Resignation, the oil crisis/crises, disasters like the Three Mile Island incident, and the aforementioned Iranian Revolution. The 70's were probably one of the U.S.'s worst decades (though previous periods were WAY worse, like the Civil War, WWI, the Great Depression, WWII, etc.) politically. Even with the U.S. celebrating its 200-year anniversary, all the bad stuff happening then stuck out pretty badly. See? I remember the stuff I learned in U.S. history in high school!