There are no blatant references to the Narrator's name in the movie. The source material, the 1996 novel of the same name, has the main character refer to himself in the third person as "Joe's arbitrary organ" a few times, but this could be a red-herring as it may refer to an article the character read. Of course both the article being about a Joe and the character being named Joe is possible, but not confirmed. The Author has stated that even he doesn't know his real name. It does have a scene where the character shows Marla "proof" that he isn't Tyler, via his driver's license. It does not have Tyler's name, but considering the Narrator is an unreliable narrator, there is always the doubt his license is also fake (or was replaced by the Tyler persona).
The movie on the other hand does not have Joe. It has Jack, as in "Jack's Smirking Revenge":
The Narrator uses "Jack's arbitrary organ" phrasing multiple times. Most people will rely on just that to call the character Jack, but others point out that the Narrator reads a Reader's Digest article about the first person view of internal organs, called "I Am Jack's ____". (FYI: The articles are real, and old, 1970s old. The series was Joe and Jill, not Jack like in the movie.)
Aside from the article reference, the movie script definitively names the character as "Jack".
People were always asking me, did I
know Tyler Durden.
INT. SOCIAL ROOM - TOP FLOOR OF HIGH RISE -- NIGHT
TYLER has one arm around Jack's shoulder; the other hand
holds a HANDGUN with the barrel lodged in JACK'S MOUTH.
Tyler is sitting in Jack's lap.
They are both sweating and disheveled, both around 30; Tyler
is blond, handsome; and Jack, brunette, is appealing in a
dry sort of way. Tyler looks at his watch.
The change from the script to the credits is really a stylistic approach, as the character's real name is indifferent to the moral of the story, that "Jack" is an everyman. His name is not important to the story.
But again, the Narrator must have a given name in-universe, as he has led a "normal" life up until he develops Dissociative Identity Disorder.
In the movie, there is only one reliable hint of the Narrator's name. A blink and you miss it view of his paycheck. (Will update with screenshot when possible) The prop was later auctioned off, with a Certificate of Authenticity. Screen used prop of The Narrator's real name, Jack Moore:
I am Jack's 38-dollar-per-hour desk job.