From Stanford Athletics FAQs.

"Stanford traditions
"Why is Stanford’s nickname the Cardinal?
One of the first things admitted students learn about Stanford is that the team name is the singular Cardinal, not Cardinals, and that it refers to the color, not the bird. The short answer for “why?” is that the 1972 trustees and president decided that would be the team’s nickname."

"For many years, the teams had gone by the nickname “Indians.” In 1972, North American indigenous people, particularly those in the San Francisco Bay Area, were protesting their treatment by the majority culture. Some, but not all of these indigenous people felt that that nicknaming athletic teams “Indians” was demeaning. One such person led a successful campaign to remove the “Indian” as Stanford’s mascot. It should be noted that this graduate student received his undergraduate degree from the University of California at Berkeley."

"After the trustees agreed to change the athletic moniker, the student body held an election to decide on a new name. The name that won was “Robber Barons”; however, then president Donald Kennedy said he felt that name was disrespectful to the University’s founder, railroad magnate Leland Stanford. Other names that garnered votes in the election included Sequoias and Thunderchickens."

"Kennedy discovered an old Stanford Daily article that referred to the team as “Cardinals” or “Cardinal,” after the color of their uniforms. Since he also happens to be a Harvard alumnus, and Harvard uses a color – Crimson – as its team name, he liked the idea, and made it the official moniker."