The cut-off age is 14, however, if a younger child is present and able to sign their name, they are allowed to do so.
Quotes from the Adjudicator's Field Manual Chapter 71:
Signature : Be certain the application is properly signed by the applicant if he or she is 14 years of age or older; otherwise by his parent or guardian.
AND
When the oath of allegiance is administered to the applicant, he should stand with his right hand raised if physically able to do so. Thereafter, he shall subscribe to the oath on the reverse of the duplicate certificate and the officer shall complete the jurat, showing the date the oath was administered, the officer’s signature, and title. In the case of applicants who are unable to understand the meaning of the oath because of youth or mental incapacity, the officer shall endorse the oath portion of the duplicate certificate with the words “oath waived”, followed by his initials.
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When the certificate is delivered personally to the applicant (or his parent or guardian), the receipt shall be taken on the back of the duplicate certificate below the oath. Otherwise, delivery shall be made by certified mail with return receipt requested, in which event, the reverse of the duplicate certificate shall be noted to reflect the mailing and the date thereof.
The duplicate certificate shall be stapled or otherwise carefully fastened on top of the record on the left-hand side of the file folder with the left edge of the certificate at the top of the file. Moreover, since the only receipt for a mailed document will be the return receipt, hold the file until the return receipt is received, and to thereafter fasten the return receipt to the reverse surface of the duplicate Certificate.
The certificate shall only be mailed to the addressee in the U.S., as that term is defined in the Act. The envelope containing the certificate shall be marked DO NOT FORWARD--RETURN RECEIPT REQUESTED--DELIVER TO ADDRESSEE ONLY. Delivery of the certificate by mail shall not be attempted if, in spite of these restrictions, the certified article may be delivered to other than the addressee or to him/her at a place other than in the United States (for example, to the addressee's order or at an APO overseas addr ess.)