I always took that to mean that he was named Bea Jay (with initials “B.J.”)
Through the Soap and much of the Benson years, we wondered what Benson’s first name was. Well, it was Benson. When he ran for Lt. Governor, it was reveal his last name as “Dubois.”
As I recall, Hawkeye sneaked a peek at his service record, and found that he was officially listed as B.J. “He didn’t even tell the Army!”
Unless you believe the episode The Zygon Inversion, in which he says that his name is Basil.
My personal headcanon is that the Doctor’s real, honest-to-goodness birth name is, in fact, “Basil Who.” I would love for this to become canon, just for the amount of fanboy rage it would be sure to inspire. 
Or any of the episodes where he introduces himself as “John Smith”. I mean, yeah, we hear that and of course routinely brush off what seems like the laziest possible alias from some kind of space-alien trickster; but what if, in decade after decade, the guy’s just been blandly putting an honest answer out there?
No. I think it was John Hurt’s War Doctor, in The Day of the Doctor, who said that he chose that name. Also, in “Forests of the Dead” Ten tells River that there was only one time that he would, or even could, tell her his name.
I do like the thought that his/her name is actually “John Smith”.
There’s that classic murder mystery, Deduce You Say, where [del]Holmes and Watson[/del] Homes and Watkins are pursuing the Shropshire Slasher.
“Your name?”
“Shropshire Slasher.”
“Occupation?”
“Shropshire Slasher.”
Related:
“Who are you? How did you get in here?”
“I’m a locksmith, and I’m a locksmith.”
Goes to show you, never evah marry a Cartwright!
I vaguely recall a scene from a movie where a guy is begging a stripper to tell him her real name. She keeps coyly accepting the money he offers and repeating that it’s “Jane,” which seems like she’s toying with him, but turns out to be true. I had to Google for it, but apparently I am more or less correctly recalling the awful 2004 movie Closer.
Well, there’s this one guy on the team, Who’s on first…
I was in a karate class once and a new kid joined. The teacher introduced him as Mr. New and I thought “Oh, good, verbal hazing. He doesn’t even get a name. That always works out well.” It took another couple classes before I found out that his last name was actually New.
In Angel, Cordelia meets one of Angel’s friend, a guy named Gunn, which she assumed was a street tag but no, that’s his name.
The email address was “a.friend” and it was a plot point that the person wasn’t just “a friend,” but was Arthur Friend.
How about “Honey” Huan, Duke’s admirer in Doonesbury?
In the film Withnail & I, Paul McGann’s character is never addressed by name. He’s listed as “& I” in the credits. However, his name is known because at one point he receives a telegram, addressed to “Marwood”.
As for The Man With No Name…
Her actual name is Ching Huan. She became known as “Honey” because that’s what Duke called her. It’s not her real name.
The character’s name was Eric. “Hoss” means horse. His nickname refers to his size. Maybe it was more of a dream sequence than an actual flashback?
No. “Hoss” does not mean horse. Here’s the episode Journey Remembered (sequence starting about 34:00).
I remember watching that episode when it was broadcast, and being surprised that “Hoss” didn’t actually mean “horse.” But the name probably stuck because people assumed it did. (Actually, I think it was a retcon, and the name originally was supposed to be a variant on “horse.” But officially it wasn’t, according to the show.)
This one kinda-sorta fits the OP:
On the sitcom New Girl, there was a character named Winston Bishop, and another who was known only by his last name - Schmidt - until late in the final season.
Turns out, Schmidt’s first name was also Winston. Years earlier, he had lost at Rock/Paper/Scissors to determine who had to give up his name to avoid confusion.
In the epic poem Beowulf the character of Grendel’s mother is famously without a name.
Or is she?
Battaglia and others have pointed out that the name of the Earth/chthonic goddess Gefion is also given in the poem. Both have similar traits. Etc.
From the opening of The Prisoner. Many folks have interpreted that last line as “You are, Number Six” with the meaning that Number Six is now Number One.