The DS-260 presented in the question does not exactly track the appearance of the DS-260 exemplar cited in the comments.
The DS-260 presented in the question, however, contains a reference to the signer's Selective Service System form DS-1810. This form refers to the SSS status of the applicant or the applicant's family members who are included in the application and will be entering the United States. It does not (necessarily) refer to the "preparer" of the application.
Therefore, the signature fields you reference should be completed with the applicant's passport number. "The form is mine" wording in the question suggests the son (the writer of the question) is the applicant. If that's correct, then the son's passport number should be entered.