There is no conclusive answer to this at this point. Chell's last name has been redacted from the records, however, the frequent references to adoption and some of GLaDOS's lines have led people to speculate that she may have been Caroline's daughter, and that Caroline and Cave may have been romantically involved.
It appears to me that Cave died about 10 years before Chell was born so he can't be her father. But Caroline is with all likelihood Chell's mother who probably left her baby on a doorstep to protect her from whet she was about to become. Figures. Anyway, if you translate the lyrics of the turret opera into English one of the lines is "my child" so that might refer to Caroline revealing her identity. I just hope Chell speaks Italian.