Nixon wins in 1960 with no southern strategy. George Wallace and the Dixiecrats subsequently take over the Democrats. Republicans more or less become the party of the left whilst the Dems are to the right (at least socially, economically i'm guessing both are towards the centre).
Going back to the discussion on the previous page, 1960 was too late for such a major shift. The black vote was already heavily Democratic, and Northern liberals were
way more influential in the party than the Dixiecrats, who simply weren't numerous enough to make a difference.
George Wallace did come close surprisingly close to winning the Democratic nomination in 1972 and 1976, but for that he had to rebrand himself as a "racial moderate" and seek the support of the Northern working class. Him being the nominee at some point would be an interesting scenario due to the sheer clash between him and the liberal activist types, but this site is way too prone to a great man theory of presidential nominees, wherein Wallace would immediately overtake the Democratic Party and reshape it into his own socially conservative image.
It'd be like if someone in a "Gore wins 2000" TL asserted that if Bush, who supported immigration reform and got a relatively large amount of the Latino vote, had become president, then the Republicans would become pro-immigration, the Hispanic vote would turn safe Republican, and Democrats would be forced to become the anti-immigrant populism.
Socio-political trends are more powerful than any one individual person. Nixon passing the Civil Rights Act instead of Johnson would have made some ripples, but I don't see a major direct difference. The majority of the Democratic Party - the Hubert Humphreys, the Ken Galbraiths, the Eugene McCarthys - would still vote for civil rights and continue to defend them against the Dixiecrat minority. The conservative Republicans, who had comprised the party base since 1932, would continue doing their damnest to claw the Eastern Establishment out of power, and they would eventually find an ally in the Dixiecrats, who were linking segregationism to "small government" values and were becoming increasingly likely to side with the Republicans on other social and economic issues anyway. "Liberal Republicans vs Populist Democrats" scenarios aren't completely implausible, but they require a change far more intricate than some guy winning one election.
And, of course, this thread completely throws the butterfly effect out the window. Most of the discussion seems to be "how can we get to a point wherein the Republican Party's policies match those that Obama supported as president?" (e.g. an individual mandate-based universal healthcare policy), which is the complete wrong way to go about this. If we're continuing with the strategy of changing the Republican Party to suit Obama rather than changing Obama to suit the Republican Party and we're assuming that any major changes to the American political system won't have any effect on Obama's early life, the question is "how can we get to a point wherein the Republican Party is one that Obama would want to join when he first became political active?" (presumably during his college years). And that requires a much deeper dive into his early life and and his mentality during that period than I have time or energy to do right now.