Now I know, before all the variously named fans of Star Trek start freaking out at me, that Kirk's middle name was established as 'Tiberius' before this film. Tony Keen has written an excellent article about it here, where he explains that the name was originally established in the Animated Series in the 70s, and it was also mentioned in the novelisation of Star Trek: The Motion Picture. Tony quotes the relevant section from the novel, which is rather intriguing in itself - Kirk, as narrator (though to be honest, the writing doesn't sound terribly Kirk-like to me) says he is 'forever tired of explaining' that he was given the middle name 'Tiberius' because the Roman Emperor fascinated his grandfather Samuel. This is pretty bizarre, since no matter how fascinated his grandfather was, it seems unlikely that he would want his grandson named after him - that's like naming your offspring 'Caligula'. Why the writers of the Animated Series gave him that name in the first place is an even bigger mystery - presumably they just wanted to give him a really awful secret middle name for fun, like Endeavour Morse, and thought 'Tiberius' sounded suitably exotic.
But the Animated Series isn't canon, and isn't repeated anywhere near as often as the other series (I don't think I've ever seen a single episode) and I'm not sure whether the movie novelisations are supposed to be canon or not (if they are, that makes Saavik doing David Marcus canon, which is fairly bizarre) but they're also going to reach a smaller number of people than the series itself. For those of us who mostly watch the five main series and the films, Kirk's middle name is revealed in The Undiscovered Country, and although the Roman emperor is not name-checked, the connection is, I suspect, intended to add something to the scene.
The Undiscovered Country (taken from Hamlet and claimed to mean 'the future', though in its Shakespearian context it means 'death/the afterlife', which should have told them something really) sees the Federation and the Klingon Empire make peace following a disaster at the Klingons' main energy producing moon. Following an interesting dinner with the Klingon Chancellor ('note to the galley - Romulan Ale no longer to be served at diplomatic functions') the Chancellor is assassinated and Kirk and McCoy are arrested and (very quickly) put on trial for the killing.
The case for the prosecution is brought by the wonderful, wonderful Christopher Plummer, delivering lines of Shakespeare in Klingon with great gusto. (Their council for the defence is Worf, entertainingly enough, supposedly menat to be his grandad or something). He accuses McCoy of either being incompetent or deliberately allowing the Chancellor to die, and Kirk of plotting the whole thing because he hates Klingons. In the middle of his interrogation, Plummer's General Chang addresses Kirk by his full name, heard by many people in the audience for the first time, with all the emphasis on the middle name - 'James, TIBERIUS, Kirk!'
Presumably, for anyone who isn't familiar with Roman history, this just sounds like a funny name. But for anyone who knows who it refers to - an emperor notorious for sexual perversions and for allowing one of his subordinates to run the Empire and murder anyone who ticked him off - it has an extra frisson to it. Chang is accusing Kirk of murder and reflecting on his history of insubordination, which resulted, among other things, in his demotion from Admiral to Captain. There's no suggestion that Kirk is, in any way, like Tiberius, but somehow the fact that he happens to have the same middle name as a particularly nasty emperor gives Chang another petty little weapon to throw at him. The reveal is great - hilarious and wonderfully delivered - and poor Kirk just starts to look smaller and smaller. (He looks pretty tiny already, due to the massive Klingon courtroom that seems to be presided over by a character from Labyrinth).
The new movie has kept the name - The Undiscovered Country is one of the best Star Trek movies, well known and loved, so with that, the Animated Series and the novel, it's pretty well established now. (Yes, I know, Kirk was also supposed to be born in Iowa, but it's pretty easy to imagine that his mother went into early labour when her ship was attacked). Here, the name is explained as an unfortunate family name - Mum suggests it as a first name and Dad says no, cause it's terrible. It's mostly just a funny name, though extra terrible if you know the history. By the time we reach the scene in which teen Kirk steals his step-father's car, Kirk is seemingly proud of the name - it marks him out more than 'James' or 'Kirk' and more importantly, in the new mythology, it comes from his now late father's side of the family. Chang hurled it at him as an accusation, but now it's a badge of honour. So we might be hearing it a bit more often from now on.