What is the quality that he possesses that makes him unique among the Edain, and why is he, of all Men, the one that is going to kill Morgoth in the Dagor Dagorath?
Turin Turambar was raised by Elves in his childhood. Such fostering enabled him to learn such "Elven teachings". Keeping in mind that the Elves were wiser and more powerful than Men, these teachings, which include; sword craft, woods craft and possibly many other survival skills, made him a powerful character.
Even so, such a fostering isn't exclusive to him only. What else makes him unique?
The curse Morgoth laid upon him.
In the tale of Turin, who named himself Turambar 'Master of Fate', the curse of Morgoth seems to be seen as power unleashed to work evil, seeking out its victims; so the fallen Vala himself is said to fear that Turin 'would grow to such a power that the curse that he had laid upon him would become void, and he would escape the doom that had been designed for him'.
The Children of Hurin
The curse is precisely the reason why Turin is chosen to be the Edain to kill Morgoth in the Dagor Dagorath. His whole life was under the shadow of Morgoth; all his decisions and counsel go awry, he kills friend and foe alike, and eventually kills himself because he unknowingly
weds and beds his own sister.
He is known as the tragic hero of Tolkien's Legendarium.
For all the wrongs Morgoth did to his father, his mother, his sisters, and himself most of all, Turin earned the right to avenge the Children of Hurin, and even all Edain, by dealing the blow that kills Morgoth in The Final Battle.
Then shall the Last Battle be gathered on the fields of Valinor. In that day, Tulkas shall strive with Morgoth, and on his right hand shall be Eönwë, and on his left Túrin Turambar, son of Húrin, returning from the Doom of Men at the ending of the world; and the black sword of Túrin shall deal unto Morgoth his death and final end; and so shall the Children of Húrin and all fallen Men be avenged."