Excellent question. And the answer, we think, is just as interesting.
Yes, there was plenty of Confederate influence in Nevada during the Civil War. But the "rebels" were definitely in the minority and they were actively pursued and persecuted by Union sympathizers and government functionaries in the majority.
The discovery in 1859 of the Comstock Lode, around which Virginia City and Carson City grew up, was an important factor in the establishment of Nevada as a United States territory on March 2, 1861, when it was cleaved off of the gigantic Utah Territory. Abraham Lincoln, president at the time, was greatly interested in ensuring that the Comstock’s riches in silver and gold would be controlled by the Union and not the Confederacy.
So when the Civil War began a month later, on April 12, 1861, Nevada was a federal territory, a part of the Union, though like everywhere in the country at the time, the divisions were deep and often martial.
For example, on June 6, 1861, two dozen U.S. Army troops from nearby Fort Churchill were ordered to investigate rumors of Confederate agents in some of the mining camps. The troops confiscated weapons from suspected citizens in Carson City and Silver City, then marched to Virginia City to check out a rumor of the flying of a Confederate flag.
The Army officers believed there was a secret organization of Southern sympathizers in Virginia City. Arriving there, they formed two companies of fifty volunteers each, issued the confiscated arms to them, and had them swear an oath to protect the Union and suppress any rebellious actions. A number of "rebels" were taken into custody, hobbled with balls and chain, transported back to Fort Churchill, and put to hard labor.
The first territorial governor was James Warren Nye, a former New York City police commissioner, whose sympathies were firmly on the Northern side. He, too, went to some lengths to squash demonstrations in Nevada in support of the Confederate States of America.
In an interesting aside, in 1864, as the War Between the States was winding down and Lincoln was up for reelection, he needed new states and their electoral votes to defeat General John C. Fremont, the Radical Republican candidate, and General George B. McClellan, a Democrat, both of whom he’d earlier relieved of their military commands.
Nevada was one of three territories invited into the Union shortly before the presidential election of 1864. Nebraska's constitutional convention voted against statehood, and Colorado’s voters rejected the proposed state constitution. In Nevada, however, Union sympathizers were so eager to gain statehood that as soon as the state constitution was ratified, by an overwhelming majority (88%) on October 31, 1864 (still celebrated as Nevada Day), they transmitted the entire document to the U.S. Congress by telegraph, fearing that sending it by train, it might not arrive on time for the election on November 7. The Nevada State Constitution, reportedly, set a record as the largest and costliest transmission ever by telegraph. The president declared Nevada a state the same day.
Thus, Nevada was the only territory to support Lincoln. Ironically, it turned out that Nevada’s votes were superfluous. Fremont dropped out of the race and Lincoln won in a relatively easy election over McClellan.
As for the nickname of the University of Nevada-Las Vegas’s athletic teams, "Rebels" was chosen in 1954 when the Nevada Board of Regents established the Southern Regional Division of the University of Nevada (known then as Nevada Southern), breaking off from University of Nevada-Reno, which was more than 75 years old at the time. Students adopted the Rebel name and mascot (the Confederate with his long gun) to facilitate the school’s emergence from the shadow of the much bigger and older school to the north.
The name "Runnin’ Rebels" was coined in 1974 by UNLV’s sports-information director at the time, but it refers only to the UNLV men's basketball team.