Beam Me Up, Scotty!: Unless the player is fast with button presses, Navi doesn't normally say "Listen!" right after "Hey!". Her automatic halts of the gameplay doesn't include the "Hey" at all.
Development Gag: Phantom Ganon's use of Portal Pictures for his fight's second phase is a remnant of how the game was originally going to use them for traveling through the world (a la Super Mario 64) instead of having Hyrule Field.
When using the ocarina, the ability to shift the pitch with the analog stick. This isn't mentioned anywhere in the instruction manual, and has little effect on gameplay, but it's a nice little touch that makes it closer to the real thing. Some people have used it as an actual instrument, even. Observe.
At least one Player's Guide for the original release had a pull out cardboard page with all the songs (plus empty space for a scarecrow's song notation) on one side, and instructions on how to play the BGM of Kakariko Village on the ingame Ocarina using these tools. These were included in the official Nintendo Power players guide, though the song notations were placed in a centerfold rather than a removable page.
There are also portraits of Mario, Peach, Luigi, Bowser, and Yoshi viewable from the castle's courtyard. These were replaced with a New Super Mario Bros.-like scene in 3D.
I Knew It: Hyrule Historia eventually confirmed the popular fan theory that Rauru and Kaepora Gaebora were the same person.
Killer App: Managed to be this for both the N64 and the 3DS.
Old Shame: Miyamoto and Aonuma have both apologized for the Water Temple, as it went beyond fairly challenging to simply maddening due to a combination of the bad item UI for the Iron Boots and the water level adjusting mechanic. The Master Quest (aka Hard Mode) expansion's redesign of the dungeon actual made it easier than the original. In the 3DS remake, it ended up being the only dungeon to experience any major changes aside from the graphical facelift - icons and luminous wall markings were added throughout the dungeon to make remembering out where to go to change the water level easier. The re-assigment of the boots to button icons also made it less infuriating.
The Red Stapler: Over ten years later, replicas of the Ocarina of Time and other Zelda-styled flutes can still be found at ren faires and the like.
Throw It In: The fishing minigame. According to an Iwata Asks, the fishing mechanic started off as one designer's way of entertaining himself during the making of the game (basically the equivalent of playing Solitaire at work). It was quickly discovered, and others were impressed enough that they shoved the fishing minigame into Lake Hylia, admitting that they had trouble finding an area in the game to place it. It helped that the previous game in the series also included a fishing minigame, though it was much less involved.
Finding the Triforce, defrosting Zora's Domain within the game, beating The Marathon Man. The former is based on pre-release trailers showing Link finding the entire Triforce; the second only happens during the ending, and the latter is impossible; the programmers couldn't figure out a good prize for beating him, so the Marathon Man always beats you "by one second". Even if you somehow cheat the race to finish in 0 seconds or less.
Also, the Temple of Light as a hidden dungeon. This one was proven false, though it's theorized that it was in the beta version, but was Dummied Out.
The game was originally going to be released on the ill-fated 64DD add on, but was moved to cartridge once it became clear that the 64DD might not even see a US release.
Mahito Yokota (main composer for Super Mario Galaxy) originally wanted to remaster the soundtrack for orchestra for the 3DS version, and even prepared several tracks, but Koji Kondo abruptly asked him to "recreate the N64 sound." In the end, only one orchestral track made it into the game.
Ocarina was originally going to be mostly in first-person, with the camera going into third-person during combat. This was quickly dropped due to the guy who made Link's model not standing the thought of not seeing Link all the time. The idea of a first person adventure game with some third-person moments was used for a different franchise.
Originally, there was no Child Link. Just Adult Link for the whole game.
The game was envisioned to have some Dating Sim elements; this is why there are so many plot important women and so few men. The idea was dropped due to the storyline making all of Link's potential choices unavailable in some fashion.
The final battle with Ganon was originally going to be a Colossus Climb, and was dropped for the same reasons as the first person mode. Though it would later be implemented in Super Mario Galaxy which shared a good chunk of the same development team.
The Forest Temple and the Water Temple respectively were originally planned as The Wind Temple and The Ice Temple. The medallion designs reflect this. It's more apparent in Ganon's Tower, as the forest section has fans all over, and the water section is completely ice and snow.
The game runs on a heavily modified version of the Super Mario 64 engine. Originally it was going to be much more similar, down even to using the iconic Portal Pictures. This idea was considered because the developers weren't sure at first if they were going to be able to pull off a large, open world on the Nintendo 64. Fortunately, they were able to get it to work, resulting in the game we have now. The Phantom Ganon boss battle is a Development Gag of this idea.
Some fans seem to be very insistent that each Iron Knuckle enemy is in canon a brainwashed Gerudo girl, citing the fact that Nabooru's face appears inside all of them (this can be seen by clipping through them in 1st-person view). The polygon model has her face inside because one Iron Knuckle is revealed in-game to actually be her inside a suit of armor, and there was no point in the artists rendering a separate model for the other Iron Knuckles (who in canon are empty suits animated by evil magic) without it since it's never meant to be exposed. It should be noted that the face is gone on all other Iron Knuckles in the 3DS version.
Many fans who ascribe to the "fourth Triforce piece" theory point to the extra triangle at the bottom of the Hylian Shield as evidence, even after the game's creators went out of their way to Joss the theory. Note The Hylian Shield in all later games lacks the lower triangle, instead giving the red Loftwing on the design a long tail. Aonuma stated at one point it was a move made just to get people to stop asking about it. Despite this, the possibility of an extra piece to fill in the middle space is used heavily in fan fiction and by people who only really played Ocarina of Time. This was part of what finally set off the creator of the website Gannon Banned.