How does the meaning of Montresor's name in "The Cask of Amontillado" relate to what "makes" him seek revenge upon Fortunato?
In French, Montresor translates to my treasure, and, once we ascertain what Montresor treasures or values highly, it is not hard to figure out what he'll be willing to do to protect that thing that he values so much. Early in the story, Montresor states that he has borne a "thousand" injuries at Fortunato's hands, but when Fortunato "insult[ed]" him, that was the last straw. He feels that he must avenge himself; however, he says, "a wrong is unredressed [...] when the avenger fails to make himself felt as such to him who has done the wrong." Thus, Montresor must take his revenge, but it is vital to him that his enemy knows that it is he who is responsible for that enemy's downfall. This sounds a lot like pride.
Further, when Montresor reveals his family motto to Fortunato -- a Latin phrase that translates to You will not harm me with impunity -- as well as the fact that the "'Montresors [...] were a great and numerous family,'" Montresor's pride comes into clearer focus. He not only has intense personal pride but familial pride as well, and his once-large family apparently prided itself on getting the last word, so to speak, in any dispute. It also seems possible that Montresor is even more protective of his family pride since the family is no longer as "great and numerous" as it once was.
What Montresor treasures, then, is his pride, in all its forms. It is the quality that compels him to seek the revenge he does, and it is the same quality in Fortunato that Montresor exploits in order to exact his revenge. Anyone who treasures his own pride so deeply will be unable to let any personal slight go, and Fortunato has evidently heaped them on. Therefore, Montresor's need to avenge his wounded pride is what makes him commit such a terrible deed.
While most studies of irony focus on the ironic meaning of Fortunato's name in "The Cask of Amontillado," the symbolism in Montresor's name is often overlooked. In the Romance languages, Montresor's name literally means "to show fate." At the beginning of the story, Montresor states why he desires Fortunato's downfall. He tells the reader that
"the thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult, I vowed revenge."
This is the only clue that Montresor offers in regards to his motive in killing Fortunato. If a family feud truly existed between the two men, it was so long ago that Montresor doesn't mention specifics, or so one-sided that Fortunato does not even recognize the danger that Montresor presents. Montresor seems to be keeping track of Fortunato's supposed insults against him and sees it as his fate to cause Fortunato to meet his fate.
Montresor is obviously another one of Poe's unreliable narrators, and Poe might have chosen the name to demonstrate that mentally unstable people often view themselves as more important than they are--Montresor must lead another to his fate just like the narrator of "The Black Cat" must destroy anything that bothers him and then "outsmart" the police.