Were states’ rights relevant to the American Civil War?

Absolutely.
The issue was as relevant to the South as it was to the North.
In January 1850 United States Sen. Henry Clay of Kentucky introduced a compromise bill to settle a four-year dispute over whether the territories of California and New Mexico—acquired by America in her victory over Mexico in 1848—would be annexed into the Union as slave or free. The dispute began shortly after war against Mexico had been declared in 1846 because Texas had been granted statehood in December 1845, and North and South quarreled over territory claimed by Texas that appeared to interfere with the 36-degree longitude, 30-degree latitude parallel North established by the Missouri Compromise of 1820. That legislation established that all territory above the 36-30 demarcation line was declared free, while territory below it was declared slave.
Sen. Clay sought to end the growing sectional conflict over territories annexed by Mexico. By September 1850, the bill was ratified in Congress with the following provisions: California was to be admitted to the Union as a free state. The organized territories of Utah and New Mexico were allowed to exercise popular sovereignty in deciding the issue of slave or free. Texas dropped its claim to land north of the 32nd parallel North and west of the 103 West meridian (known as unorganized territory, which included parts of New Mexico, Oklahoma and Colorado) in exchange for the federal government assuming the former Texas Republic’s debts. In addition, the slave trade was abolished in the District of Columbia, but Congress also strengthened the Fugitive Slave Act (FSA). The latter authorized and empowered slave owners to claim their property in free states, regardless of individual state personal liberty laws.
This last provision, the Fugitive Slave Act, brought the sectional conflict between North and South over slavery to an entirely new level of bitterness, contention and outright hostility.
Abolitionists and other antislavery societies refused to recognize this federal edict, and encouraged others to resist it as well. In fact, entire municipalities and states joined in the resistance, forbidding slave catchers and/or federal marshals from using local or state facilities or from receiving assistance of local or state authorities in the capture of fugitive slaves.
Many northerners, in fact, insisted that the federal government recognize state-sanctioned personal liberty laws, the aim of which were to protect free blacks, freedmen and fugitive slaves from capture by guaranteeing them due process of law. These laws included, among other things, requiring the right to a jury trial and an appeal before a slave could be extradited back to his or her master; forbidding the use of local jails; and denying cooperation from local or state officials to aid in the capture of fugitives. It was a legal way around the FSA without invoking the doctrine of nullification, which had been used before unsuccessfully by South Carolina in 1832 over tariffs.
As such, the issue of states’ rights was very much relevant to many Northerners in antebellum America, especially during the turbulent 1850s.
States’ rights, though, has historically been associated with the antebellum American South and southern rights.
Venerable U.S. Sen. John C. Calhoun of South Carolina brought the issue of southern states’ rights to the fore in the early 1820s. He realized the need for southern states to maintain fair representation in Congress, since the majority of the American population resided in the North and House seats were designated by districts according to population. The greater the population, the more seats in the House a state and, therefore, region possessed.
Calhoun further recognized that representation in the Senate was the South’s best opportunity for equal representation in Congress, because each state was allotted two senate seats, regardless of the population of that state. As such, Calhoun began to lobby passionately for territorial expansion west of the Mississippi with the aim of settling much of that territory under pro-slavery—and, therefore, pro-South—conditions and attaining statehood as slave states. This strategy, Calhoun surmised, would ensure the South held a strong voice in Congress for decades to come, no matter how loud, powerful or influential the antislavery voices became.
Therefore, he lobbied on behalf of southern “states’ rights”—and state interests—for fair representation in Washington, D.C.
The “states’ rights” doctrine was based on Congressional representation, first and foremost, but it also included the right of southern state interests—namely slavery—to expand and settle into the new territories. States' rights also included the assurance of individual property rights—again, namely slavery—to the people of southern states settling or expanding into the new territories.
Popular reference to “states’ rights” has to do with the constitutional right of states to be granted certain political powers, which maintain local sovereignty and autonomy within the national republic, as directed by the Tenth Amendment to the U.S. Constitution.
But state sovereignty really had less to do with the “states’ rights” of the antebellum American South than the protection of slave property did.
Southern states’ rights reached at the very core of debate and disagreement over the issue of slavery in America. It boiled down to whether individuals and, to a broader extent, states had the right to exercise their Constitutional guarantee of property (Fourth and Fifth Amendments to the U.S. Constitution) and the movement thereof in the new territories west of the Mississippi River; territories which were not yet organized and not yet represented or recognized in Congress.
Southern interests contended that slave property was protected, even in unorganized territories claimed by the United States, while northern interests disputed this claim. The Southern rights movement further argued that it was unconstitutional for the federal government to arbitrarily declare slavery illegal in the new territories, because it denied property rights of southern settlers, thereby violating the U.S. Constitution.
Interests in the North, on the other hand, countered by saying that because free labor would be directly and negatively affected by slavery in the new territories, the issue was one of commerce, which falls within the Congressional powers of Article I, Section 8 of the Constitution. Many in the North opposed the expansion of slavery for economic reasons; chiefly that it provided unfair competition to free labor. Others in the North who opposed slavery, though, did so exclusively on moral grounds; upon which many abolitionists and other antislavery societies staked a higher authority than even the Constitution.
And while the question of slavery was by far the most significant issue of the antebellum era, states’ rights was no less significant in relation to the slavery institution and, thus, not irrelevant to the cause of the American Civil War.