I think it shows that Lamech was filled with self-importance. The reference for this "avenging" is :Gen 4:19 Then Lamech took for himself two wives: the name of one was Adah, and the name of the second was Zillah. 20 And Adah bore Jabal. He was the father of those who dwell in tents and have livestock. 21 His brother’s name was Jubal. He was the father of all those who play the harp and flute. 22 And as for Zillah, she also bore Tubal-Cain, an instructor of every craftsman in bronze and iron. And the sister of Tubal-Cain was Naamah.
23 Then Lamech said to his wives:
“Adah and Zillah, hear my voice;
Wives of Lamech, listen to my speech!
For I have killed a man for wounding me,
Even a young man for hurting me.
24 If Cain shall be avenged sevenfold,
Then Lamech seventy-sevenfold.”
Lamech is clearly self-important by referring to himself in third person to his wives.
So Lamech is saying that his murderous activity is 11 times worse than the murder committed by his great-great-great grandfather?
What is his reasoning?
Gen. 4:14 Surely You have driven me out this day from the face of the ground; I shall be hidden from Your face; I shall be a fugitive and a vagabond on the earth, and it will happen that anyone who finds me will kill me.”
15 And the Lord said to him, “Therefore, whoever kills Cain, vengeance shall be taken on him sevenfold.” And the Lord set a mark on Cain, lest anyone finding him should kill him.
Lamech for whatever reason wanted even more protection for himself by claiming the same curse on anyone who would try to avenge the murder he committed. Except Lamech said, Well if great-great-great grand dad was protected with 7 times vengeance, then surely I should be protected not just 7 times, but 77 times the vengeance.