In a statement released Friday through Shakespeare's Globe Theatre, Rylance and his family said "our beloved daughter and sister Nataasha passed away of unsuspected natural causes early on Sunday morning."
Van Kampen, a 28-year-old filmmaker, was the daughter of Rylance's wife, composer Claire van Kampen.
Tony winner Rylance, regarded as one of Britain's finest actors, was due to recite from "The Tempest" during the July 27 opening ceremony.
Director Danny Boyle has said the ceremony's theme, "Isles of Wonder," was inspired by Shakespeare's play about castaways on an enchanted island.
Rylance is still due to star in upcoming productions of "Richard III" and "Twelfth Night" at the Globe.