The narrator remarks that the name “William Wilson” is fake; he chooses it for the purposes of this story, but it isn’t actually his real name. This deception is interesting in and of itself – and you can read more about it in “Character Analysis.” For the moment, let’s think about why he chose this particular name.
First, you’ve got the doubling going on inside the name itself; “William Wilson” is a tad redundant, in that “WILLiam sounds a lot like “WILson.” Then you can go nuts with name etymology, if that’s your thing. “William” originally combined two words, “will” (as in desire”) and “protection.” When you put them together, you get “will protection.” Or the narrator has willed himself some protection. If you like the theory that the second William Wilson is the first’s conscience, then you could think of him as protection for the narrator; a conscience protects us from doing things that are wrong. Now, if the narrator has willed his own protection, it hints that the narrator has deliberately willed the existence of his double, and that this double in some way protects him – from himself, or at least from his most ungovernable passions. That’s what a conscience is for, in any case. Lastly, the origin of “Wilson” is “the son of William.” You could look this up on an etymology site, or you could just think about the word: “Will-son,” or “son of Will.”
Of course, the second William Wilson’s physical likeness to the narrator William Wilson is one of the other big indications that they may be two halves of the same person. It is this likeness of feature, not the identical name, that so terrifies the narrator in the early half of the story, and that is the primary reason he is unable to look his double in the face for the rest of the tale.