"States' Rights" is historical revisionism by ex-Confederates. This is all part of the "Lost Cause" mythos that was created by white Southerners after the war was lost. Even during Reconstruction, but especially afterwards, white Southerners began to construct a false narrative about the "War of Northern Aggression", casting themselves as the noble, chivalrous "defenders" of their homes and rights against the numerically and industrially superior Northern army. The "right" they are talking about is, of course, the right to own slaves. While the North did not start the war, and did not initially prosecute it to free black slaves, but rather to restore the Union, the Southern States rebelled to preserve that right, and that right alone. All else they did was secondary to the preservation of chattel slavery.

This narrative was included in books and articles circulated in the South, and in public education, as well as works of popular culture and civic societies like the Daughters of the American Revolution Confederacy and the {spits} Ku Klux Klan.

All of these institutions, writings, and attitudes contributed to a re-creation of Southern society along ante-bellum lines, with black people on the bottom and white people on top. It was a conscious repudiation of the efforts of racial equality put forth in the Constitutional amendments and Reconstruction-era policies enforced by Northern politicians.

This false narrative permeates much of the South's most cherished cultural touchstones, Gone With The Wind being the best example. "Birth of a Nation", a seminal work of film-making in the 1920s and arguably the first American "epic" film, also expounds on this theme and re-enforced it for an entire new generation.

While the term itself was in use before the Civil War it was rare and not very popular. It exploded in popularity starting with the 1870s and 1880s when the first Lost Cause histories started to be published.

There is more than enough evidence from the time period to show that the main cause for the Civil War (from the Southern perspective anyway) was the issue of slavery/white supremacy.

Ta-Nehsi Coates has a fantastic piece in the Atlantic titled What This Cruel War Was Over. I'm going to take some quotes from his piece, as he's already done the work and there's no need to duplicate efforts.

South Carolina's statement on why to go to war:

A geographical line has been drawn across the Union, and all the States north of that line have united in the election of a man to the high office of President of the United States, whose opinions and purposes are hostile to slavery. He is to be entrusted with the administration of the common Government, because he has declared that that “Government cannot endure permanently half slave, half free,” and that the public mind must rest in the belief that slavery is in the course of ultimate extinction. This sectional combination for the submersion of the Constitution, has been aided in some of the States by elevating to citizenship, persons who, by the supreme law of the land, are incapable of becoming citizens; and their votes have been used to inaugurate a new policy, hostile to the South, and destructive of its beliefs and safety.

Mississippi's:

Our position is thoroughly identified with the institution of slavery—the greatest material interest of the world. Its labor supplies the product which constitutes by far the largest and most important portions of commerce of the earth.

Louisiana:

As a separate republic, Louisiana remembers too well the whisperings of European diplomacy for the abolition of slavery in the times of an­nexation not to be apprehensive of bolder demonstrations from the same quarter and the North in this country. The people of the slave holding States are bound together by the same necessity and determination to preserve African slavery.

Texas:

in this free government all white men are and of right ought to be entitled to equal civil and political rights; that the servitude of the African race, as existing in these States, is mutually beneficial to both bond and free, and is abundantly authorized and justified by the experience of mankind, and the revealed will of the Almighty Creator, as recognized by all Christian nations

Jefferson Davis:

You too know, that among us, white men have an equality resulting from a presence of a lower caste, which cannot exist where white men fill the position here occupied by the servile race. The mechanic who comes among us, employing the less intellectual labor of the African, takes the position which only a master-workman occupies where all the mechanics are white, and therefore it is that our mechanics hold their position of absolute equality among us.

James Henry Hammond (of the famous "Cotton is King" speech):

We do not think that whites should be slaves either by law or necessity. Our slaves are black, of another and inferior race. The status in which we have placed them is an elevation. They are elevated from the condition in which God first created them, by being made our slaves. None of that race on the whole face of the globe can be compared with the slaves of the South. They are happy, content, unaspiring, and utterly incapable, from intellectual weakness, ever to give us any trouble by their aspirations

Mississippi Senator Gallatin Brown in 1858:

I would spread the blessings of slavery, like the religion of our Divine Master, to the uttermost ends of the earth, and rebellious and wicked as the Yankees have been, I would even extend it to them.

From the Florida secession convention in 1861:

At the South, and with our People of course, slavery is the element of all value, and a destruction of that destroys all that is property. This party, now soon to take possession of the powers of the Government, is sectional, irresponsible to us, and driven on by an infuriated fanatical madness that defies all opposition, must inevitably destroy every vestige or right growing out of property in slaves.

Regarding the issue of "States rights". I've seen it pop up a few times in pre Civil War documents, but not many. Oddly enough one time was by Joseph Smith (founder of the Mormons) in conversation with some close supporters. This was later reprinted in the Mormon-owned newspaper The Millennial Star

The States rights doctrine are what feed mobs. They are a dead carcass, a stink . . .

This was in reference to Calhoun's doctrine that the states had the power to abrogate and nullify any Federal decree and the Federal government had no power to interfere in state business. Smith had sought protection for Mormons being persecuted by mobs (which were sometimes instigated by state officials), and the Calhoun doctrine basically said that the Federal government was powerless to interfere--thus this reaction.

Calhoun used the phrase in 1834:

"My aim is fixed. It is no less than to turn back the Government to where it commenced its operations in 1789.. to take a fresh start, a new departure,on the State Rights Republican tack."

And there's a flyer I found calling for a "State Rights Convention" (a phrase close enough to "states rights") to be held in Charleston, S.C. on July 1st, 1830.

Edit:

/u/turtleeatingalderman probably has some comments saved talking about this subject. /u/Irishfafnir too.