What is the significance of Lamech's boast? Lamech said to his wives,
   “Adah and Zillah, listen to me;
    wives of Lamech, hear my words.
   I have killed a man for wounding me,
    a young man for injuring me.
   If Cain is avenged seven times, 
    then Lamech seventy-seven times.”
  (Genesis 4:23-24, NIV)


What is the significance of Lamech's boast? What role does it play in the Genesis narrative?
Lamech killed some younger man. Lamech represents the fifth and last generation mentioned of the Cainites, who by that time were probably a large multitude occupying cities. Lamech represents the culmination of those that turned away from the promise of the Messiah to Eve, in contrast to the Sethites which were the early church.

What we have in Lamech's short poem to his wives, whose names in Hebrew mean 'ornament' and 'shade', is the height of violent boasting.  He essentially say's 'I will kill (and have) who I want and anyone who avenges it shall suffer!'  Grabbing beautiful wives (a polygamy of ornament and 'shade', possibly cast from the lovely flowing hair by his second wife) and having children that were committed to establishing crafts and skills and building cities, implies that in addition to the violence of this people, they were vain, lustful and seeking permanent settlement into the pleasures and ambitions of this world.  The Sethites on the other hand were more like visitors to the world, looking for a better life by faith. Basically Lamech's great boasts, lusts and violence was the early version of Heathenism later represented so well by Greece and Rome.

Note: In my answer I have mostly summarized points learned from Alfred Edersheim's 'The Bible History, Old Testament  Volume I ' but the idea of two races developing with different agendas early in Genesis is common to most Bible Commentaries.