What might have happened if the North had not lost interest in Southern Reconstruction after 1870?

The pressing issue facing both North and South after the Civil War was how to incorporate a population of newly freed slaves into Southern society. More progressive members of Congress, usually members of the Republican Party, wanted to build an integrated South that granted full civil rights to blacks. As a result of this goal, they clashed with the more conservative wing of their own party and with the Democrats.
In the late 1860s, events came to a head. President Johnson, who supported states' right, allowed Southern states to pass repressive legislation meant to ensure that blacks would provide low-cost labor to whites. In 1866, Northern congressmen passed a Civil Rights Bill that said all people born in the United States were citizens with equal rights before the law. They also passed legislation extending the life of the Freedman's Bureau, a federal agency tasked with helping Southern blacks. Unfortunately, Johnson vetoed both laws. This helped lead to his impeachment and to an override of his veto of the the Civil Rights Act.
For a brief period in the 1860s and even into the 1870s, blacks were elected to Congress and had hopes of achieving integration and greater equality in the South. This would have still been a racist society, but it would have been a society significantly less openly racist than what emerged.
Instead, the readmittance of Southern states into the Union, the loss of interest in Reconstruction, political dealmaking, and Northern white fears of too much social change led to the situation that in fact occurred. The Ku Klux Klan was allowed to operate freely and terrorized blacks. Under the guise of states' rights, former Confederate states enacted repressive legislation that denied blacks voting rights, enacted segregation, and sharply curtailed full black equality. 
If the North had insisted on implementing the early vision of a South in which black rights were fully protected, the gains of Civil Rights movement of the 1960s could have potentially been enacted 100 years earlier. Blacks would have been allowed to vote freely, and acts of terror, such as lynchings, would have been fairly prosecuted. Additionally, much black suffering and exploitation would have been avoided.  If the United States had dealt with its institutional racism during Reconstruction, we would almost certainly be much further along in establishing racial equality in this country today.