"His name is Luke."
Translation:Han heter Luke.
First of all, you should inflect the noun, so it would be 'navnet', as the possessive comes after. Secondly, 'sin' are wrong for two reasons: 1. 'navn' is neuter, so it should be 'sitt', 2. 'sin' refers to a subject, but in this case 'Luke' is not a subject, so it should be 'hans'. Using 'sin/sitt' is wrong.
"Navnet hans er Luke" is accepted.
This actually translates to "Hans navn er Luke", not "Han heter Luke". BTW: "Han skrev ned navnet sitt" = "He wrote down his name" referring to his own name "Han skrev ned navnet hans" = "He wrote down his name", referring to somebody else's name (masculine) "Han skrev ned navnet hennes" = "He wrote down her name" (the other person being, of course, feminine).