Twitch Plays Pokémon, a crowdsourced Let's Play that takes inputs from an IRC channel. Despite the chaotic nature of it, somehow, within 16 days of continuous playing, they managed to beat the Elite Four and Blue, and managed to catch Zapdos, a legendary Pokémon, to boot!
Sleeper Hit: Nintendo of Japan actually said It Will Never Catch On while releasing Japanese Red and Green and writing it off as a loss. It didn't top the sales charts, but it kept selling steady in a market where 80% of sales are made in the first two weeks. Game Freak made a few tweaks, released Japanese Blue and Nintendo got their cash cow.
Urban Legend of Zelda: There's a rumor that's been floating around since these games came out about a hacked version known as Pokémon Black (not to be confused with Pokemon Black And White), found on a black-painted cartridge. The player is given a Pokémon known as 'Ghost' instead of one of the starters, and it cannot be put into the PC. It attacks opponents with Curse, and if you return to an area after beating a trainer, it's indicated they died as there's a tombstone present. After beating the game, the player character's Pokémon are all gone and the sprite is swapped for an old man sprite. Going back to Pallet Town results in the character being attacked by the spirits of the Pokemon they killed on their way to beating the game, with the character using Struggle until they die. Its existence is questioned, but a ROM like this does exist online. Article here
A rumor has always been around that when the games were first released in Japan, loads of little children got ill and committed suicide because of the Lavender Town music. Some people thought this was why the music was subtly changed to remove some of the high pitched notes in the American remakes. In reality, the music was changed because people in Japan had been getting headaches from the shrillness of it.
The Missing No (or Missing Number) has had a number of rumors around it, some said catching it would damage your cartridge others say that it would erase your data.
There's a truck sprite that only appears once in the game. The only way to see it is to go way out of your way to avoid triggering a certain plot event that closes off the area and returning when you can travel on water. People believed that it had to have been put there for a reason, leading to hundreds of rumors about this thing. As it turns out, it's just a truck.
Then there's all the rumors about how to catch Mew....
Not to mention to rumored existence of additional secret Pokémon, from the infamous "Pikablu" to the Poké Gods, and the incredibly difficult and convoluted ways to get them to appear in the game.
Speaking of Vermilion City, the S.S. Anne was said to return there after the player beat the Elite Four so many times.
Averted in that exploitation of how the game engine processes data will allow you to encounter any monster of your choosing...including Mew.
There once was a female protagonist◊. She was excluded presumably due to a lack of space for her data and reused for FireRed and LeafGreen as "Leaf". Her design was used for Blue (known as Green in the US) Pokémon Special.
Cutting Pokémon for space is also why Mewtwo is in the Pokedex before Mew; there originally wasn't enough space until the debug tools were removed late into production. Had there been enough room it is conceivable that Mew would have been placed before Mewtwo in the Pokedex.
During development, the Sevii Islands were considered. But, due to a lack of cartridge space and time limits, they were dropped. They were added in the remakes FireRed and LeafGreen.
Trivia for Yellow:
Recursive Adaptation: It's a game based on The Anime of the Game, and in addition to some Canon Immigrants, there are some touch-ups on sprites and Pokémon used in Gym battles: Brock and Misty wear their respective outfits from the anime (in the original games, Brock was shirtless and Misty wore a bikini), and Lt. Surge only uses Raichu in battle, which is what he also did in the anime.note He used 3 Pokemon in Red and Blue, and the remakes and later games follow this instead.