I think the Zelda fandom is a unique one in which we don't need a name to classify ourselves with. Even with concept names such as "Links" or "Hylians", it really doesn't make sense as there are games that don't really have the land of Hyrule as the basis of the overworld such as Majora's Mask or Skyward Sword.
There are two kinds of fans of Zelda-
1. People who where Zelda shirts and necklaces but little to know connection to the games
2. People who have the Zelda games and are on sites like Zelda Dungeon. Also, a Skyward Sword Wii Mote is an automatic sign of a hardcore fan in my book, I don't have a name for us, but we are, at the very least, hardcore fans.
Idk, we could call ourselves Zeldites or something?
Not necessarily true, for there are people who may not have all of the games or merchandise in the Zelda franchise, but their knowledge in the series is what should be d Not everyone is going to have the same opinions of every game in the series. Some people may consider a certain the best in the series, while others might believe that is actually the worst. This can be applied towards the many concepts that makes the franchise for what it is such as plot, characters, game mechanics, theories, etc. It's for that reason that specific names to name people within the franchise wouldn't work as the term may not be relevant to what they like. It's also for that same reason why I believe that the Zelda franchise doesn't need a name for their fans and people who enjoy Zelda. That's what we are at the core,
Zelda fans, and that's one of the greatest titles that any fan could have.