Flip Flop of God: Though some of this is also the fault of Word of Dante, there's been conflicting reasons for why Sora and Roxas are able to dual-wield Keyblades. The current official explanation is that it's because he has Ven's heart inside him, and Roxas is able to do so because he's Sora's Nobody. This is a direct flip of what fans had previously assumed—that Sora got the power from absorbing Roxas, and Roxas in turn got the power from absorbing Xion.
Hey, It's That Voice!: Ye gods, this. One of the games' most acclaimed features, in fact.
In Re: Chain of Memories, Kairi and Namine, who are essentially two halves of the same whole, are voiced by Alyson Stoner and Meghan Jette Martin respectively. Those two also played archenemies in a rather popular movie. Make of that what you will.
No Export for You: Kingdom Hearts owns this trope (unfortunately). Neither Americans nor Europeans got their hands on Kingdom Hearts: Final Mix or Kingdom Hearts 2: Final Mix+, making everyone mad and further mucking up the narrative.
Which was quickly countered by Birth By Sleep: Final Mix, with Nomura being evasive when asked of a release outside of Japan.
The Other Darrin: Quite a few. A bad example being Mandy Moore replaced by the absolutely wooden Mena Suvari. A good example would be Lance Bass being replaced by George Newbern (who made it clear that the line is "Descend, Heartless Angel", not "Sin Harvest" as Bass' rushed delivery had many believing.)
Corey Burton is The Other Darrin to a few of characters, most notably Di Z. He played MCP and Sark in II, although the original actor did want to come back but had schedule conflicts.
Geoffrey Rush, Captain Barbossa's actor in the movies, wasn't able to initially reprise his role in the second game. He did reprise the role for the Australian dub.
Providing a good example for why this trope is used, Kathryn Beaumont is the one voice actor from Walt's era who reprises her roles (Alice and Wendy). About fifty years later, they naturally don't sound at all convincing.
Schedule Slip: Yoko Shimomura apparently has habitual problems producing her music on time. This got a whole lot worse during production for the handheld trilogy (coded, 358/2 Days, and Birth by Sleep), during which she was also composing for Final Fantasy Versus XIII, leading to Tsuyoshi Sekito and Takeharu Ishimoto composing a good deal of Birth By Sleep's music.
What Could Have Been: The game was originally supposed to be Mickey-themed, but there was a disagreement on whether Mickey or Donald should be the protagonist.
Further in development, Sora was half lion and had more of a chain-saw blade then a keyblade.
Chernabog was to have been the original Big Bad, the embodiment of all darkness, taking inspiration from American Gods with the final battle being a gradual climb up the slopes of Bald Mountain to fight him. He was turned into a surprise boss from console limitations and Executive Meddling.
The final battle with Xemnas in KH2 was supposed be quite different, with Xemnas melding into a giant Transformer-esque creature◊ after absorbing power from Kingdom Hearts and, upon defeat, transform into this◊ distorted version. This idea was nixed and replaced with the dragon-fortress/armored Xemnas and black-and-white coat Xemnas battles.
Variants of Behemoths and Wyverns (from the first game) were included in the 1000 Heartless Battle in Kingdom Hearts II.
Goofy and Donald were originally supposed to battle in their classic costumes, according to trailers, but these were replaced with Nomura's designs.
The Pridelands were originally going to be featured in the first game, but the engine couldn't handle the quadrupedal movement, so Simba was included as a summon instead. After changing the game engine, the world managed to make it into Kingdom Hearts II.
358/2 Days had a Pinnochio world which was Dummied Out. It would have included Gepetto, Foulfellow and Gideon, and Human!Pinocchio, and would have a What Measure Is a Non-Human?-centered plot and a focus on circus scenes. It was cut due to space limitations.
The outside walls of Hollow Bastion/Radiant Garden were originally going to be playable in Kingdom Hearts II.
Kairi's role in Hollow Bastion in the first game was going to be larger.
An odd example is a scene from the Beta trailer of the first game, which has Riku at the top of Big Ben. What's odd is that, despite never appearing in the finished game, it is still referenced in later games as canon.
Both Hades and Megara were set to appear in 358/2 Days.
Rikku was originally supposed to have been with Leon, but the production team decided it'd be too confusing with Riku and Rikku, so she was replaced with Yuffie, who also got the outfit Rikku would have worn◊.
Vincent Valentine was planned to feature in the first game, but was replaced by Cloud, a more recognizable and iconic character.
A younger version of Irvine Kinneas was also supposed to be in Destiny Islands, but was replaced with Wakka (also more recognizable, although perhaps because of Kingdom Hearts).
Both Leon's and Cloud's pasts were going to be touched upon in Birth by Sleep, but as the world already had several members of Organization XIII appearing, it was decided that the world would have been too overloaded with characters.
Marluxia was originally envisioned as a woman, but was changed to a man because of the Unfortunate Implications of having two women attempt to overthrow a mostly male organization and falling flat.
Axel was supposed to be killed by Roxas at the end of the prologue in Kingdom Hearts II, but was such a fan-favorite that he was brought back for the rest of the game and given a Heroic Sacrifice.
The members of Organization XIII who died in Chain of Memories (Vexen, Lexaeus, Zexion, Larxene, and Marluxia) were going to be bosses in the Olympus Coliseum in KH2, but were cut. The Data-Battles of Final Mix are a remnant of this.
There was planned to be one more Ansem!Riku battle at the End of the World. It was very nearly completed but Dummied Out in the final release.
The original scenario involved Sora saving the Princesses of Heart with the assistance of their respective princes.
Miscellaneous trivia:
"Destati" is apparently considered cursed by the production team due to odd incidents—the worst being the entire building losing power—occurring whenever the vocal track was inserted into a piece of background music.