Paul McCartney’s mother was given the name Mary. More fully, she was Mary Patricia McCartney, née Mohin, making “Mary” the correct canonical answer to the question as asked.
Mary was born into an Irish-descended Liverpool family and qualified as a nurse. She later worked as a visiting midwife, and her income helped the family move to 20 Forthlin Road in Allerton, the house that became closely associated with Paul’s early life and songwriting.
Paul was born at Walton Hospital in Liverpool on 18 June 1942. His younger brother Peter Michael—later known professionally as Mike McGear—was also born at the same hospital. Music was encouraged at home by their father, Jim, but Mary’s work was particularly important to the family’s finances.
Mary died on 31 October 1956, aged 47, from an embolism following surgery for breast cancer. Paul was 14. Her death became one of the emotional experiences linking him with John Lennon, whose mother Julia died less than two years later. Paul and Linda McCartney later named their first daughter Mary in her memory.