Your son can be admitted to the US solely based on his green card. He does not need his passport to be admitted to the US. He will probably need his passport to exit your country of origin, though.
Does it take more than one year to obtain a passport in your country? Maybe there is some other way? My country issues passports in about 6 to 7 months (if applied through the embassy in Washington DC), but they will issue a temporary one within two to three weeks, and you can still keep your old passport (invalidated however). If I were to fly to my home country to apply there I would get the passport in 10 business days.
Does it take more than one year to obtain a passport in your country? Maybe there is some other way? My country issues passports in about 6 to 7 months (if applied through the embassy in Washington DC), but they will issue a temporary one within two to three weeks, and you can still keep your old passport (invalidated however). If I were to fly to my home country to apply there I would get the passport in 10 business days.