Jayhawkers were the Union's most notorious war criminals, born from the "Bleeding Kansas" segment of the Civil War. Kansas, being pro-Union, fought their battles mostly in Missouri. But Jayhawkers did not go by the guidelines of war, and they - along with the Confederacy equivalent in Missouri, Quantrill's Raiders (which included the psychopath known as Bloody Bill Anderson - who had grievances as well; one of his sisters was killed and the other was maimed after poor housing in the prison they were in collapsed on top of them - and a young Jesse James) were especially brutal.
Both sides did things that made Sherman's notorious "March to Atlanta" seem like a picnic for little children. Those two sides fought all along Missouri, but the Jayhawkers were very much notorious for massacring Missourian farmers, raping Missourian women, and burning and looting towns in Missouri under a guise of them being "Confederacy sympathizers" (which a lot of those people weren't). Quantrill's Raiders went into Kansas and massacred people and raped women there. Both sides would've been tried under the Geneva Convention because what they did was beyond the pale.
Meanwhile, the Missouri Tigers were named after a militia regiment in Columbia, Missouri - volunteers who only wanted to protect their town from Jayhawkers and Quantrill's Raiders, along with other detachments from both Union and Confederacy. They successfully did so, and Columbia was amongst the few towns around that area to not be sacked.
There are many "Civil War" rivalries going about, but no rivalry in all of sports has as much hatred and history as Missouri/Kansas.
Tl;dr, Mizzou named their sports team after the defenders of their city who committed zero war crimes. Meanwhile, KU names their sports teams after a bunch of murderers and rapists.