Just the other week I developed a proposal for the Wikibox of the Week thread and made a brief list of elections for it that just about meets the criteria of the Challange, and could be easily changed to allow for such a thing.
By various means (I chose to have FDR Live longer) one can develop a strain of Social Democracy in the Democratic Party, a Second New Deal/Second Bill of Rights, a Fair Deal enacted, avoiding Kennedy, Dixiecrat and McGovern type factions in Democrats while supporting the triumph of Stevenson/Harriman Johnson/Humphrey line of thought can lead to a serious triumph of Social Democracy, alot of people in the late 1960's before the Vietnam quagmire and the Conservative blow back thought it was possible. Avoid Vietnam and the Social Left Front can go on unbroken.
With this agenda after Eisenhower it might very well be impossible for the Republicans to ever gain traction again, deal at the same time with divisions in the Conservative movement caused my McCarthy, George Wallace, the Segregationists, Barry Goldwater etc and you can break the party. I had the GOP in my scenario die in the early 1970's but it could go on theoretically even after its defeats.
In the face of massive social engineering, the repeal of part of the bill of rights, a massive cradle-to-grave welfare state and US foreign policy being on a big Nationbuilding/Peacekeeping bent the Decline of the GOP and the Left-shift of the Democrats leaves alot of political minds left in the void, and thus in the 1970's a push for a smaller government unites under the Libertarians.
In the event of the Timeline the change has to come in 1972 where HHH, Ed Muskie, or another figure representing the Left of the New Deal Coaliton, who's not George McGovern or Eugene McCarthy takes office. Then the cycle can get back on track, With Nixon representing a final blip on the GOP chances radar. After this canidate beats Nixon off again in 1976, you start seeing that same shift I described, so that by the 1990's the GOP is dead in the water, and in the 1980's the Libertarians rise up as the new opposition party (Having also to beat off an attempt by the Reform Party to do the same).