Banned in China: The series was banned in Saudi Arabia, Thailand and United Arab Emirates.
Development Hell: In 2009, Sony Pictures registered several domain names pertaining to possible Grand Theft Auto movie. However, no new information was released after that.
Dueling Games: With the Driver series initially, thanks to some not-so-friendly jabs that III took towards that series. In the IV era, the Saints Row series became its main rival, with ads for Saints Row 2 mocking IV's perceived stonefaced seriousness.
Fan Nickname: 'Flipstalker' for the Landstalker in the 3D era games for reasons that are apparent for anyone who has tried doing sharp turns with it.
Gratuitous German: The Securicars are run by a "Gruppe Sechs" which translate to "Group Six" (not a really interesting factoid, eh?).
No Dub for You: A rare Western example. None of the games have non-English voice tracks, and when they are released internationally, they are subtitled instead of dubbed.
Oh God, San Andreas had a ton of these. Perhaps the most prolific were the rumors that Bigfoot (or a Sasquatch, or a Yeti) roamed the forests, and that there were "ghost cars" that drove around in the countryside. The former turned out to have been faked with judicious use of Photoshop, while the latter was a result of abandoned, wrecked cars spawning atop hills and then rolling down them (quick note: when cars spawn, they don't have their parking brake on). Other rumors included zombies (mostly the result of a building in San Fierro that housed a company called "Zombocorp"), UFOs, the Loch Ness Monster, Jaws, and a masked slasher who ran around the woods with a chainsaw.
IV also had a few. There were rumors that Lola, a prostitute who appeared on some of the promo art, could be found in-game, that a creature called Ratman could be found lurking the subways, and that an abandoned factory is haunted.
A full list of features, missions, characters, etc. that didn't make the final cut of their respective games can be found here.
Around the time of production on III and Vice City, Rockstar Games reportedly considered setting a GTA game in Tokyo, going so far as to scout the city for ideas. The idea fell apart for two reasons: the road network was seen as too convoluted to make for a fun open-world driving game, and the culture gap would've meant completely changing so much of the series' style that the game would've been virtually unrecognizable as a GTA game.