Bloodline runs from father's name

No.

(I would offer a correction, but it's not clear to me what you're trying to say. A bloodline can run through the mother or the father, but not from anyone's name.)

Thanks Khoff--and yes you are right I should have given some context.

I have noticed (I might be wrong) that for generations it's the father's name that 'runs down', never the mother's. It's the practice in our culture at least. Am I able to get my point across now?

Thanks,

Tom

Okay, you might want to say something like "It's the father's surname that is handed down from generation to generation."