When I sent my minor children abroad alone, I always gave them notarized letters of consent. No official at any American or European border ever asked to see any documentation beyond their passport.
I would treat this situation as the girl traveling alone -- having an adult "with her" in some official capacity would make the situation more complicated -- and if I did persist with the letter, add the godfather under "contact information".
I would also consider giving a medical power of attorney to the godparent, just in case of either real medical emergency or some over-eager border guard questioning a man traveling with an unrelated teen-aged girl.