I was going over the history of the Republican party prior to Lincoln, and it looks like some of the tenants of the party were the same as now, such as Free Market Labor, however I'm interested as to when States Rights became a forefront issue of the party since the Republican party spearheaded the Northern effort to effectively stop states from having the right to secede. I'm very interested in how the parties essentially traded some issues like this, and the historical context.
I'm going to rely mainly on The Dixiecrat Revolt and the End of the Solid South, 1932-1968 to help answer this one.
During the 1948 Democratic National Convention, Harry Truman was re-nominated as the candidate of the Democratic Party, and adopted a plank in the party platform in favor of civil rights. In response, Southern delegates walked out of the convention, and formed the States Rights Democratic Party (also known as the Dixiecrats, and who had as their symbol a variant of the Confederate battle flag), who put forward their own presidential candidate - Senator Strom Thurmond. You can read the Dixiecrat party platform here.
The Dixiecrat strategy revolved around the "Solid South" - those former Confederate states which had regularly voted for Democrats since the end of the Civil War, and which were most opposed to de-segregation and most supportive of states' rights over federal authority. While the Dixiecrats were successful in drawing votes away from the Democratic party, and Thurmond actually carried the states of Alabama, Louisiana, Mississippi, and South Carolina, Truman still won the election.
The Dixiecrats as a break-away party did not last beyond the 1948 election, but the heightened political tensions as de-segregation gained ground led to the development of the "Southern Strategy" - that is, turning the Southern Democrats who believed in states' rights and segregation to the Republican party. In the 1964 political campaign, Barry Goldwater ran as a conservative Republican opposed to the Civil Rights Act of 1964 - he was defeated, but his pro-segregation platform was a deliberate call to Southern voters; in 1964 Strom Thurmond even joined the Republican party.
The actual implementation of the "Southern strategy" was in Nixon's presidential campaigns of 1968 and 1972, which culminated in '72 with the election of Nixon with mass appeal, north and south. The actual "Southern strategy" was explicitly explained by Lee Atwater in a 1981 interview, contained in Southern Politics in the 1990s:
Atwater: As to the whole Southern strategy that Harry Dent and others put together in 1968, opposition to the Voting Rights Act would have been a central part of keeping the South. Now [Reagan] doesn't have to do that. All you have to do to keep the South is for Reagan to run in place on the issues he's campaigned on since 1964 . . . and that's fiscal conservatism, balancing the budget, cut taxes, you know, the whole cluster...
Questioner: But the fact is, isn't it, that Reagan does get to the Wallace voter and to the racist side of the Wallace voter by doing away with legal services, by cutting down on food stamps?
Atwater: You start out in 1954 by saying, "Nigger, nigger, nigger." By 1968 you can't say "nigger" — that hurts you. Backfires. So you say stuff like forced busing, states' rights and all that stuff. You're getting so abstract now [that] you're talking about cutting taxes, and all these things you're talking about are totally economic things and a byproduct of them is [that] blacks get hurt worse than whites. And subconsciously maybe that is part of it. I'm not saying that. But I'm saying that if it is getting that abstract, and that coded, that we are doing away with the racial problem one way or the other. You follow me — because obviously sitting around saying, "We want to cut this," is much more abstract than even the busing thing, and a hell of a lot more abstract than "Nigger, nigger."
Famously, during his 1980 presidential campaign Ronald Reagan made a speech where he said:
I still believe the answer to any problem lies with the people. I believe in states' rights and I believe in people doing as much as they can for themselves at the community level and at the private level. I believe we have distorted the balance of our government today by giving powers that were never intended to be given in the Constitution to that federal establishment.
And with the election of Reagan, "state's rights" was just about ensconced in the Republican political agenda, practically a complete reversal of policy positions from when it was the party of Lincoln. The whole process was more complicated than I'm summarizing here, and de-segregation fueled other movements that tied into it, because while the presidential campaigns that looked to flip the South are perhaps the most visible, the Republican transition to "states rights" also occurred at local levels of government as well.