We just did this with my 12 year stepdaughter right after her mother naturalized. They wanted her personal appearance, her green card, her foreign birth certificate, her mom's naturalization certificate, her mom's CA State ID, her foreign passport for ID and entry stamps, proof of legal custody (which is her case was her birth certificate, might be different in other cases), and lots of proof that she was actually living here in the US with her mother. We initially supplied her health insurance papers with our address, some USCIS notices in her name at our address and tax returns showing we claimed her as a dependant. They were not satisfied with that and seemed fixated on school records to prove she had been living here so we got her school ID, attendance record, report card and enrolment record from her school. We did it at a Passport Agency with an urgent appointment for planned international travel within 14 days and for us that was a HUGE benefit since they returned all our evidence the next day, allowed us to e-mail the additional school records they requested the same day and issued the US Passport the next day. The Passport Agency kept her green card when they issued the US Passport.