Most likely is 1948, but 1980 comes up as a strong second. '48 requires President Obama to run against a gridlocked Congress, possibly facing third-party opposition from the Left, with the economy continuing to be in crisis, against a candidate who calls for the undoing of the modern post-New Deal State.1980 assumes the Republican are going to nominate a far-right candidate, and President Obama sinks due to a poor economy and personable opposition who placates public fears during the campaign.
There's no way that this is a 2004 election. The economy alone prevents this from being a 1996. If a third-party candidate enters on a fiscal angle, it could be similar to 1992.
If the Republicans nominate a far-right conservative, the election could be either 1964 or 1980. On the one hand, inflation is very low across the board, though food prices, health care, and college tuition are increasing well above the overall CPI. Politics is more polarized now, and people are deeply unhappy with the state of the economy, so 1980 is more likely than 1964.
1976 is unlikely based on the current state of the Republican field. Romney and Carter have a lot in common, since Romney has been campaigning for the past several years, as Carter did after being termed out as Georgia Governor in 1974. However, Ford had never been elected and was swamped by the anti-Nixon effect. All the same, he connected well with the American people, despite his sometimes-Radical past in the House as Minority Leader -- pushing for the impeachment of Supreme Court Justice William Douglas, for example, and had a strong chance at victory in the last few weeks if "no Soviet domination of Eastern Europe" hadn't happened.
1948 is possible. Unlike 1952, there's no chance a Republican candidate will pull an "I will go to Korea" moment and declare an end to the wars. The economic situation was also disastrous, as well as massive opposition and legislative gridlock from the Right (as well as Southern Democrats, when it came to issues like public housing, the minimum wage, and desegregation). Several New Deal blocs were going wobbly, such as African Americans, and it took the rise of leaders like Hubert Humphrey to push President Truman to make a stand on civil rights and mobilize his base. Imagine Kucinich running against Obama, and that was basically the candidacy of former VP Henry Wallace, as compared to Kennedy in '80.
- 1948: Massively unpopular incumbent, third-party challenge from the Left, ongoing (Cold) war, rage from the Right, and 'our time' triumphalism following twelve years of FDR, runs against 'do-nothing' Congress after laying out a 'special session agenda' in which Congress fails to take any action, intense country-wide campaigning, massive economic meltdown post-WWII.
- I'll add 1964: Ultra-conservative candidate crushed by popular liberal incumbent who enacted sweeping social changes to country's safety net programs with good economy.
- I'll add 1976: Unpopular incumbent, unknown outsider challenger, starts from far behind and makes a strong finish to come just short of election.
- 1980: Unpopular incumbent, hostage crisis/failed rescue, OPEC oil embargo, ultra-conservative candidate proves himself as worthy through debates after a tight summer campaign to cruise to victory in the fall, ultra-high inflation.
- 1992: Unpopular incumbent, third party challenger, just-finished war and ongoing economic recession, incumbent seen as out of touch with public (milk prices, supermarket scanner) and party ('chicken Kiev' speech, 'read my lips'/1990 Omnibus Budget Reconciliation Act), weak opposition candidate emerging from weak opposition primary field turns out to have surprising empathetic skills in connecting with the public.
- 1996: Moderately popular incumbent, post government-shutdown and takeover of House by radical opposition, opposing candidate had a public reputation for being a radical and an inside-game reputation for being a moderate, turning off both the public and party, respectively, booming economy.
- 2004: Moderately unpopular incumbent, polarized campaign, incumbent ran a 50%+1 base-heavy strategy, ongoing wars, budget deficits but reasonably strong economy, post-9/11.