Creator Breakdown: The very concept of the game and the sequence when Link is turned into Deku Link are coincidentially based on daydreams and nightmares. The stress of the deadline Eiji Aonuma had to make a Zelda game in a single year gave Aonuma nightmares, one of which involved Deku. The next day, the scene director showed him a scene he created for the game, which was just like Aonuma's nightmare; they decided to keep it in. As for the concept of the game, it was partially inspired by Yoshiaki Koizumi's daydreams, where he would wonder what it would be like if the moon suddenly fell down to Earth, and how people would react.
Executive Meddling: The game was originally supposed to be an expansion to OoT, titled "Ura Zelda". However, when the designers expressed displeasure with this idea, and the Nintendo 64DD hardware add-on that would have enabled it to be an expansion failed, they were challenged to make an entirely new adventure in 1 year.
Prop Recycling: Most of the character models are recycled from Ocarina of Time, which greatly reduced development time for this game.
Serendipity Writes the Plot: As noted above, the developers only had a year to complete the game (which is very short compared to the other Zelda games), forcing them to reuse models from Ocarina of Time while justifying this in that Majora's Mask is set in a Bizarro Universe of OoT.
Early screenshots showed the moon to have no face.
The game began development for the N64 add-on called the 64DD (or the "Nintendo 64 Disk Drive"), and was going to make use of some of the add-on's features. The game would have used a real time clock instead of the in game one, and was going to use a full 7 day week instead of three days like the final game, but these were dropped from the final product since there were difficulties trying to get these features to work on standard N64 hardware (even with the required RAM expansion pak). There was also a proposed feature regarding being able to literally buy time from a merchant, but the change to the clock system and lessening of days made the merchant seem less important.
Instead of the Fierce Deity's Mask, there was going to be a mask in the game that would have let Link turn into his adult self from Ocarina of Time (evidence of its existence can be seen in really early screenshots). This mask was scrapped for being mostly useless since all major equipment could be used by young Link this time around. Not wanting to let the "adult Link" aspect go to waste, however, the developers put in the Fierce Deity's Mask instead as a 100% Completion bonus that's required to use on the final boss in order to see the Golden Ending. Remnants of this idea include the fact that Fierce Deity Link reuses adult Link's voice clips and model (not textures), as well as an unused mask depicting Link's regular face.
Early screenshots also display various changes to textures throughout the game. A small example is that the Deku Pipes originally were colored white instead of brown, but a much bigger example is some of the wall textures for both inside and outside of Clock Town. The textures originally were a bit more surreal with weird-looking suns, but in the final game the walls textures instead got somewhat less surreal looking designs. Perhaps another very interesting change is that Link was originally going to have the Kokiri Shield from Ocarina of Time return in place of the Heroes Shield (or perhaps as an alternate starting shield), and the Kokiri Sword originally looked exactly like how it did in Ocarina.
Originally, separate icons were used in the HUD for each of Link's instruments (which change depending on which form he's in). The final game just used the Ocarina icon. This idea would be revisited for the 3DS remake, which did have separate icons for each instrument.
Early screenshots showed the Megaton Hammer or something like it as Goron Link's weapon. This was replaced by a simple punch for the final game.
One image taken from the earliest stages of development showed that all three Gorman Brothers were originally just one Ingo, and might've worked for Cremia at one point instead of having competition against her.
Before release, the game contained something called the Inspector's Mask, and you were to receive it from Mr. Toto. This mask did not make it into the finished game, and what it would have done remains unknown; however, considering Mr. Toto is familiar with Kafei's mother, and that Kafei's mask allows you to ask people where he's been, it could just have been an earlier version of that.
Shigeru Miyamoto told Famitsu that there was going to be an enhanced fishing mini-game. In the game, Link would be fishing for the legendary Jabu-Jabu—the Terminan version of the whale/fish deity from Ocarina of Time of the same name. If he managed to catch the giant fish truly bizarre things would happen, such as halting the fall of the Moon, or rupees raining from the sky. The fishing minigame would later be implemented in the remake, and it is possible to catch a rare fish that resembles a smaller Jabu-Jabu named "Lord Chapu-Chapu," though nothing spectacular happens when you do.