The female gymnast known for her "balance beam routine" during the 1996 Olympics is Shannon Miller. She was a prominent figure in gymnastics at the time and earned multiple medals, contributing significantly to the U.S. gymnastics team's success in Atlanta. Miller's performances were marked by their precision and artistic flair, showcasing her training and dedication. Her achievements helped elevate the visibility of women's gymnastics, setting a standard for future generations of athletes in the sport.