LONDON—The mother of a murdered child who became the face of a News of the World campaign to change British sex-offender laws says she was on the list of a private investigator who hacked phones for the tabloid— a development that is likely to generate more scrutiny for Rebekah Brooks, the former News Corp. executive who edited the paper at the time.

Sara Payne, in a prepared statement, said Thursday that police had found her details on a list seized from Glenn Mulcaire, the private investigator who worked for the now-closed...