Could a mother authorize a relative/friend to travel with her children (12-13 years old) from the US to another country? What are the rules and requirements?
Would an authorization letter from the mother to show at immigration enough?
You can authorize someone who is not a parent or legal guardian, and would need to verify the destination country entry and, potentially, exit requirements.
As they (and/or you) are in the United States, here is what US Customs and Border Protection recommends:
U.S. Customs and Border Protection (CBP) strongly recommends that unless the child is accompanied by both parents, the adult have a note from the child's other parent (or, in the case of a child traveling with grandparents, uncles or aunts, sisters or brothers, friends, or in groups*, a note signed by both parents) stating "I acknowledge that my wife/husband/etc. is traveling out of the country with my son/daughter/group. He/She/They has/have my permission." See our Q&A parental consent.
While CBP may not ask to see this documentation, if we do ask, and you do not have it, you may be detained until the circumstances of the child traveling without both parents can be fully assessed. If there is no second parent with legal claims to the child (deceased, sole custody, etc.) any other relevant paperwork, such as a court decision, birth certificate naming only one parent, death certificate, etc., would be useful.
Adults traveling with children should also be aware that, while the U.S. does not require this documentation, other countries may have a requirement and failure to produce notarized permission letters and/or birth certificates could result in travelers being refused entry (Canada has very strict requirements in this regard).
Additional information on children traveling alone, can be found on our website at Answer ID 449.