Question

While your name dies out,the daughter continues the lifeline of another man in another family.
What grammatical name is given to this expression ' the lifeline of another man' and what is it function?

Answer 1

Normal word order in English sentences is this:
subject / verb / object or complement

http://www.chompchomp.com/terms.htm
Look those terms up here and then let us know what you think. Hint: ignore the dependent clause ("While your name dies out")

Answer 2

Personafication