Some Northern states exercised their states' rights by declining to vigorously enforce the federal Fugitive Slave Act.
Southern states, upset about this, seceded and formed their own federal government, under whose constitution the states were prohibited from making their own decisions about slavery.
So the war was about states' rights.
The Union was the states' rights side.
Point of order. That particular "right" was seceded when they ratified the constitution. The Constitution to which they agreed, contains a clause that REQUIRES fugitive slaves to be returned to their masters.
A state cannot exercise a "right" they signed away. The Northern states were REQUIRED by the US constitution to return fugitive slaves, and they deliberately BROKE that constitutional law. Repeatedly.