Yes and no. It's traditionally first born after the father's parent. The middle name thing became more common during the forties and fifties, the period of assimilation, when people said they weren't naming their kids Rocco, Nunzio, Carmella or Rosalia, etc. Their were those who named after the grand parent but used a middle name that they would call the child; something like Guido James and the kid would only be called James. Except of course when Mom was angry, then every name was used to call the child, which indicated he was in serious trouble. The grandparents were usually satisfied if they knew their name was on the birth certificate.