What is Spock's full, allegedly unpronounceable name? (And no, I don't mean Dr. Benjamin McLane Spock. I mean the son of Sarek.)
Spock's full name is S'chn T'gai Spock.
Spock's full name was revealed in TOS novel: Ishmael. In TOS episode: "This Side of Paradise", Spock said that his full name was unpronounceable to Humans.
It is revealed in a TOS novel called Ishmael.
Apart from this reference, and the name being deemed unpronounceable in a TOS episode, there is no real proof of his name ever being said in a canonical reference (to my knowledge).
According to Memory Alpha:
Jane Wyatt, who played Spock's mother Amanda Grayson, was once asked by fans at a convention what Spock's first name was. She replied, perhaps jokingly, "Harold".  However, the question itself was flawed, since the episode "Journey to Babel" makes it clear that "Spock" is Spock's personal name.
On the other hand, Spock's family name has never been established in canon. In the episode "This Side of Paradise", Leila Kalomi says to Spock, "You never told me if you had another name," to which he replies, "You couldn't pronounce it." D.C. Fontana – who was considered the "Vulcan expert" of the TOS behind-the-scenes staff and who created such details as the fact that Spock's father was an ambassador and his mother a school teacher – revealed, in an issue of the fanzine Spockanalia, that she had intended his family name to be "Xtmprsqzntwlfd", but since this is unpronounceable, there wasn't really any way to get this said in dialogue during an episode.
In fact I remember some publication in which Spock was Spock's middle name.
His full name was something like Xantksisk Spock Xtmprsqzntwlfd. But we would have to look up whatever publication that was to find out how it was spelled.
Combining that with S'chn T'gai Spock would make a name that would be a real mouthful.