This is a really hard one. I don't know. I wouldn't want any of the Founding Fathers, both because they owned slaves and because the two-party system and a number of enduring precedents hadn't really settled down yet (Washington died before the Supreme Court decided it could overturn laws, for instance), so they'd be ill-equipped to navigate things. Some options, in chronological order (LBJ is already noted)
Pros: Abraham Lincoln. Abolished slavery. Started Reconstruction. Led us through the Civil War.
Cons: Suppression of other rights, basically only a wartime president so difficult to judge how he'd do in peacetime.
Pros: Beat the Nazis, got us out of the Great Depression.
Cons: Japanese internment camps. Didn't mind the Nazis as long as they were only killing Jews.
Pros: Antitrust, conservationist.
Cons: Big fan of the military (what do you think the big stick is?), joined a war to kill brown people and get Glory, racist as heck.
Pros: His whole 'league of nations' thing was a good idea in theory, and he did get us mostly through World War One.
Cons: Racist even for his time.
Pros: Marshall Plan. Founded NATO (probably good?) Advocated for National Health Insurance. "Fair Deal". Berlin Airlift. Recognized Israel (I'm putting that in the pros, fight me). Fought for civil rights.
Cons: Korea. McCarthyism. Lavender Scare.
????: Hiroshima and Nagasaki.
Pros: Fought Nazis. Demanded documentation of extermination camps lest people deny the Shoah in the future. Integrated the military. Interstate highway system. Massive research funding. Significantly reduced the Korean War. NASA. Civil Rights Act. Little Rock. Military-industrial complex.
Cons: Cooperated with Francoist Spain. Got us into Vietnam more. Lavender Scare. Okay, that's actually not that many. I think I'm picking him.
Richard Nixon: Okay, hear me out. First of all, the man gets a bad reputation for being corrupt. But he also was in the wrong time. Today he'd be a paragon of respectability, and fifty years before was Warren G. Harding, whose administration was incredibly corrupt. Nixon just had the misfortune to be smack-dab in the middle, when we had standards on corruption. He opened up China, got us (mostly) out of Vietnam, warmed relations with the Soviet Union, got us to the moon (okay, fine, that was mostly the people before him, but still), created OSHA, signed the Clean Air Act, helped implement major Affirmative Action programs, and generally did a lot of things he doesn't really get credit for. The cons...I don't really think I need to go into. But hey, maybe the GOP will rally around him as corrupt?
Pros: Seems to have been a decent human being and a nice guy. Signed major laws promoting mental health care. Supported a national health insurance system. Funded education more. Invited gay activists to the White House and let gay people serve in the Foreign Service. Camp David Accords.
Cons: Wasn't the greatest at managing inflation, to say the least. Iran hostage crisis.
I think I like Ike the most, though.