The item above mentions Wichita Kansas where the mothers name is. That location is her birthplace. So even if she had traveled to Kenya on a temporary visa or whatever and spit the little bastard out there he would still be a citizen of the US. (I think that's accurate). To my knowledge US woman who leave the US temporarily and have a baby have an American baby right? Even though it would appear that from the document above the kid was born actually in Honolulu. (It's kinda blurry to read so correct me if I'm wrong).
Also why would the state (Hawaii) issue a birth certificate to the grandmother of a child born in Kenya? It's not a 7-11. And it's been my experience that the states take identification and proof very seriously. They will issue documents based on facts, not just hand out birth certificates to any one who asks for one.
Immigration is easy enough in the US but it's not THAT easy.