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 Players Guide for the original release had a pull out cardboard page with all the songs (And 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 useing these tools.
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.
Good Bad Bug: To the point where there's even an entire walkthrough that utilizes almost every single one of these imaginable. You can find it here.
During the development of the 3DS remake, the developers had to go out of their way to recreate some of these bugs, because they were that damn fun!
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: Definitely not for the game itself, but Miyamoto and Aonuma have both apologized for the Water Temple, as it went beyond fairly challenging to simply maddening. The Master Quest expansion had a redesign of the Water Temple, which was easier than the original. Keep in mind, the rest of the dungeons in MQ are redesigned to be Nintendo Hard in the puzzle department. 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. (Re-assigning the boots to button icons, as they have been from Wind Waker onwards, helped too.)
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). However, 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.
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.
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. But hey, when Koji Kondo tells you to do something, you do it.
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.
Originally, the video game was envisioned to have some Dating Sim elements; this is the origin of why there are so many plot important women and so few men.
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 Nintendo64. 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.
Word of Dante: Fans seem to be annoyingly insistent that each Iron Knuckle enemy is in canon a brainwashed Gerudo girl thanks to Nabooru's face appearing inside all of them. 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.