(covers information from several alternate timelines)
"Tiberius?! You kidding me?! No, that’s the worst!"
In the alternate reality, his given name was suggested by his daughter-in-law, Winona, as a first name for their newborn son. George disagreed with the choice, and opted to have his son named after his father-in-law, James (Star Trek)
The idea was also included in the first draft script of DS9: "Trials and Tribble-ations", in which he was given the full name "Tiberius Edmund Kirk". Also in that script, Benjamin Sisko started saying, "[He] was one of the first men to climb the…" though Jadzia Dax then interrupted Sisko.
According to writer Christopher L. Bennett, author of the Star Trek: Enterprise - Rise of the Federation series of novels, Tiberius is supposed to be the son of Samuel Abraham Kirk (β), a historian that served on the USS Pioneer (β) and Valeria Williams (β), the armory officer of the USS Pioneer.
In the novelization of the Star Trek: The Motion Picture, Kirk was referred to as "grandfather Samuel" by his grandson James. The novel also reveals that James learned the swear word "Shit" from his grandfather.
According to The Autobiography of James T. Kirk, Tiberius was in his 20s when Starfleet Academy was formed and that he applied but wasn't accepted. Still wanting to get out into space, he signed on in Ordnance and supply and served on several starbases until getting assigned to Starbase 8 (β) where he met and married a nurse, Brunhilde Ann Milano (β). Their son, George Kirk was born there on December 13, 2206. Tiberius had hoped that George would go to Starfleet Academy but admission was very competitive. That dream was later fulfilled when Tiberius was awarded the Starfleet Medal of Honor for rescuing five men after Starbase 8 had suffered an explosion on the loading dock. Despite Tiberius being an enlisted man, the children of Medal of Honor winners were given high priority during the admissions process.