All There in the Manual: The novelization goes into detail on Sarah's backstory and her parents' divorce; it and other tie-ins also reveal names for the Junk Lady (Agnes) and other minor characters and locales. Curiously, Jareth's backstory is largely unrevealed, which renders that Fanfic Fuel.
Development Hell: Not the movie, but Archaia Publishing's graphic novel prequel: Announced in early 2012, initially pushed back to April 2013, now not arriving until sometime in 2014. The official explanation is that they don't want it to go out until it's perfect, and considering that it's a backstory for Jareth — the most popular character in the fandom — one can understand the hesitancy, especially given the Broken Base caused by the last attempt at an Expanded Universe with Return To Labyrinth.
Two of the Fireys were voiced by Danny John-Jules of Red Dwarf. Another was voiced by Kevin Clash, aka Elmo from Sesame Street and Baby Sinclair from Dinosaurs.
The script was written by Terry Jones, of Monty Python fame. Another acclaimed comedian/scriptwriter, Elaine May, is said to have done a substantial rewrite; Jones has said that the final third of the movie was significantly different from what he had originally come up with (starting with the peach).
Jennifer Connelly simply gets embarrassed by seeing her younger self on screen. It also doesn't help that critics ripped her performance to shreds.
What Could Have Been: Almost everything up until Sarah eating the peach was written by Terry Jones, with everything afterwards being rewritten by Henson and company before getting sent back to Jones for a final rewrite. This and other changes over the course of production resulted in quite a few alterations and omissions, some of which appeared in the tie-in books.
The Wiseman and his Hat were intended to wander in and out of the good guys' journey, dispensing occasionally useful, accidental advice.
The Fireys offered to help Sarah find the castle, but not only were they easily distracted, but they didn't actually know what a castle was.
The Junk Lady (who was going to be revealed as a disguised Jareth) was part of a whole Junk City, complete with a bar where Hoggle went to drown his sorrows after his betrayal of Sarah.
The other door with a living knocker led to a Crapsaccharine World where no one could stop laughing. (This became "The Laughing Forest" in the Japan-only Labyrinth game for the Famicom.)
The issue of Sarah's parents' divorce was dealt with more directly in early drafts. The ring Sarah gave the wiseman in the finished film was originally a gift from her mother that she was much more reluctant to part with. The novelization also goes into detail on the glamourous fellow actor who became the mother's lovernote seen only as a photo in the film, but importantly he's "played" by David Bowie — his name is Jeremy and he gave Sarah the music box.
Orignally, Sarah's plot-launching mistake was opening the door to a stranger who claimed to be the writer of the school play she was due to star in; he turned out to be Jareth, who proceeded to kidnap Toby (then called Freddie) For the Evulz. Jareth was indeed a much less charismatic, more lecherous character in early drafts — in the climax Sarah had to physically fight him off to rescue Toby, and defeated him by saying she wouldn't love him if he "were the last goblin on Earth!" This caused him to shrink into a whining goblin himself.
In the project's early stages, the story was set entirely in a Magical Land. Later the Down the Rabbit Hole structure was introduced, with the "real world" setting being the Victorian era; this was subsequently changed to The Present Day. A big reason for the changes was the filmmakers becoming aware of a similar Fairy Tale film, Legend, which went into production around the same time — in fact, the two movies ultimately shared a cinematographer.
The creators always intended for a popular musician to play Jareth and provide the songs, and while David Bowie was their first choice, another performer considered was Michael Jackson. Had they actually chosen Jackson over Bowie, the movie would have reeked of "Funny Aneurysm" Moment thanks to the accusations of child molestation brought against him in the 1990s and later.
And Henson's first choice was Sting, until his kids convinced him Bowie would have more lasting appeal.