Susan Alexandra Weaver was born in New York City on October 8, 1949. Her mother, Elizabeth Inglis (born Desiree Mary Lucy Hawkins), was a British actress and a native of Colchester, England. Weaver's father, Sylvester "Pat" Weaver Jr., was an American television executive born in Los Angeles, who served as president of NBC from 1953 to 1955 and created NBC's Today Show in 1952. Pat's brother, Winstead "Doodles" Weaver, was a comedian and contributor to Mad. Her father's American family was of Dutch, English, Scots-Irish, and Scottish ancestry.
At the age of 14, Weaver began using the name Sigourney, taking it from a minor character in The Great Gatsby. She briefly attended the Brearley School and Chapin School in New York before arriving at the Ethel Walker School (Walker's) in Simsbury, Connecticut, where she developed an early interest in performance art. One of her early roles was in a school adaptation of the poem "The Highwayman", and on another occasion she played a Rudolph Valentino character in an adaptation of The Sheik. She was also involved in theatrical productions of A Streetcar Named Desire and You Can't Take It with You during one summer at Southbury, Connecticut. Weaver reportedly reached the height of by the age of 11, which had a negative impact on her self-esteem. She recalled feeling like "a giant spider" and never having "the confidence to ever think I could act."
In 1967, shortly before turning 18, Weaver visited Israel and volunteered on a kibbutz for several months. On her return to the United States, she attended Sarah Lawrence College. After her freshman year, she transferred to Stanford University as an English major. At Stanford, Weaver was extensively involved in theater. She performed in a group named the Palo Alto Company, doing Shakespeare plays and "commedia dell'arte in a covered wagon" around the Bay Area, the nature of which she considered "outrageous". She avoided Stanford's drama department, as she believed their productions were too "stuffy" and "safe". Weaver had planned to enter Stanford's Ph.D. English program and eventually pursue a career as a writer or a journalist, but changed her mind after getting frustrated by the "deadly dry" honors courses. She eventually graduated in 1972 with a Bachelor of Arts in English. Weaver subsequently applied to Yale University's School of Drama, performing Bertolt Brecht's Saint Joan of the Stockyards at her audition, and was accepted.
Weaver admitted that she had a difficult time at Yale. She was not fond of the shows at Yale Repertory Theatre, and had little luck getting lead roles in school productions. Some acting teachers referred to her as "talentless" and advised her to stick to comedy. Weaver credited her friends such as Christopher Durang, who kept hiring her for his plays, as well as her time at the Yale Cabaret, as crucial in helping her pull through. She graduated from Yale with a Master of Fine Arts in 1974.