The Klingon Judge is Robert Easton. Yes, that Robert Easton.
So, TomRobinson, even though you are cleared of rape in the 20th century, you can't help but continue to harbor hatred and resentment to a dying race that stands against everything you hold dear? I-rony!
DRAAL expects the Federation to abide by the articles of Intersellar Law.
Blink and you'll miss it - Sulu's helmsman is the navigator of the The Red October. Give him a stopwatch and a compass, and he'll fly you through the Mutara Nebula in a starship with no windows.
Real-Life Relative: Walter Koenig's wife, Judith Levitt, is one of the Starfleet Admirals at the briefing towards the beginning of the movie.
Starfleet Admiral: Bill, are we talking about mothballing the Starfleet?
Recycled Script: Accidentally, as the writer didn't know there had been an episode of the series where Kirk fought a clone of himself. Though it does still work as twenty years of special effects advances allowed this fight to be much more convincing than the last one, which was mostly done through single shots of both Kirks.
Throw It In: General Chang was supposed to have hair, however, Christopher Plummer preferred his bald look, so General Chang became bald.
Bennett's story was seen as very risky in 1990 (recasting the roles of Captain Kirk and Mister Spock? Get outta town!), and was eventually shut down by Executive Meddling — the suits simply didn't feel that the movie going public at the time could ever accept such a radical departure. They pushed for Bennett to write a more traditional Star Trek, with all the original cast. Bennett refused to do this, and walked from the project. Nicholas Meyer, Denny Martin Flynn and Leonard Nimoy knocked together the eventual version in something of a hurry (they had to have the movie out in time for the 25th anniversary of the original series), so its probably quite surprising that the final movie turned out as well as it did.
Not just executive meddling. When told about the plan to recast, some of the original cast spread rumors at cons that it was going to be a farce along the lines of "the Jetsons crossed with Police Academy." Fans rioted and the plan was scrapped. Didn't help that Gene Roddenberry was against it as well.
Even the story they eventually went with could have been a lot deeper. In the novelization, the Klingons were responsible for a massacre on the Federation colony of Kudao (which the Klingon government claimed was a rogue action), and Chang's bird-of-prey attacked a science station on the planet Themis inside Federation space, which critically injured Kirk's love interest, Carol Marcus. Had they kept this in, it would have provided more recent context for why the Klingons were so mistrusted, and another personal reason why Kirk hates the Klingons so much— considering that David's death happened ten years or so before the events of Star Trek VI.
The original idea for the opening was to have Kirk and Spock round up the officers for their final mission. This would have shown their fates and why they were so eager to return. Scotty would be struggling to understand the cloaking device on the stolen Bird-of-Prey. Uhura would be hosting a talk radio show. Chekov would be playing chess with higher life forms, gloating about his "superior Russian strategies". McCoy would be surrounded by insufferable doctors. Kirk himself would be in bed with Dr. Carol Marcus. However, the budget would not allow for such a pricey sequence, so it was scrapped.
And if they'd gone with Saavik as the one to betray Starfleet, and if the film versions of The Wrath of Khan and The Search for Spock had followed the novelizations by having Saavik and David fall in love, it would have provided an excellent personal reason for Saavik to wish the Klingons dead.
Given that Saavik was strongly implied to be Spock's lover (in ST:III), and pregnant with his child (in the novelization of ST:IV), it would have made her betrayal and extremely public Mind Rape by her husband much more distressing than the final script's version.
Walter Koenig also submitted a story idea he was rather proud of titled In Flanders Field that would have ended with the deaths of everyone except McCoy and Spock (both of whom had already been seen alive by the time The Next Generation took place).
Homage: The speech that the warden gives Kirk and McCoy upon their entry to Rura Penthe is a paraphrase of Colonel Saito's speech to captured British P.O.W.s in The Bridge on the River Kwai. For comparison:
Colonel Saito: If you work hard, you will be treated well. But if you do not work hard, you will be punished! A word to you about escape. There is no barbed wire, no stockade, no watchtower. They are not necessary. We are an island in the jungle. Escape is impossible. You would die. Rura Penthe Warden: This is the gulag Rura Penthe. There is no stockade. No guard tower. No electronic frontier. Only a magnetic shield prevents beaming. Punishment means exile from prison, to the surface. On the surface, nothing can survive. Work well, and you will be treated well. Work badly, and you will die.