I think the standard should be 'how much would history have diverged had the other guy won' as well as how close it was. 1860 is an obvious answer; I'd also make a case for 1912 and 1936, given their potential impact on US involvement in the world wars, as well as long-term social and economic policy.
1912 didn't have that much effect. If Teddy had been President, the Lusitania incident, if it still happened, would have gotten us involved sooner, but while that would have increased Allied manpower, it wouldn't have significantly increased Allied war materiel as 1915-6 was the period we ramped up war production anyway. Maybe a year gets shaved off the length of the war and the Bolsheviks don't take power.
Given that the Cold War is a solid fifth of America's history, I'd say just that last point is a significant enough turning point. But in terms of 1912 I was talking more about the Progressive platform and the potential for the Republicans being replaced wholesale, which would be a major shift.
