You beat me to it. Some quarterback, if I remember correctly, who had cancer or some terminal disease. As he was dying, his words of encouragement to the team was "bear down", as in "don't worry about me, worry about our next opponent." He died later that season and the motto still permeates Arizona athletics.
I believe Bruschi would have been a teammate.
EDIT: Jeebus, I was so wrong it wasn't funny. My roommate in college was from Tucson, and I must have been out of my skull when he told me the story. I may have gotten him confused with Colorado's Sal Aunese right around that time. Read on:
The battle cry was created in 1926 by a popular student athlete, John "Button" Salmon. Salmon was the student body president, as well as the starting quarterback for the Wildcat football team and the catcher for the Wildcat baseball team.
The day after the first game of the 1926 football season, Salmon and three friends were involved in an automobile accident, in which their vehicle flipped over in a ravine. Although Salmon's friends were not injured, Salmon suffered a severe spinal cord injury. In the aftermath of the accident, football coach Pop McKale visited him in the hospital every day. During McKale's last visit, Salmon's last message to his teammates was, "Tell them...tell the team to bear down." John Salmon died on October 18, 1926.
The following year, the University of Arizona student body approved that "Bear Down" would be the new slogan for all Wildcat athletic teams. In 1939, the Arizona state legislature issued a decree that "Bear Down" would be the exclusive property of the University of Arizona.
