In Fairy Tail, the guild is named "Fairy Tail". While there does seem to be a range of ethnicities within the names (Natsu vs. Lucy), it just seems like an odd misspelling of "fairy tale", which is a very specific thing that seems to be totally unrelated to the guild itself.
What is the in-universe reason given for that being the name of the guild, if there is a reason given?
From Makarov's history based on Wikia:
The guild was named 'Fairy Tail' as the eternal mystery behind whether fairies had tails or not, or if they even existed, was like an eternal adventure.
So I guess the guild's (Fairy Tail's) purpose is to have a great adventure as if they are solving the mysteries about fairies.
The guild's first master Mavis loved the idea of fairies and often wondered if they had tails. This is mentioned in Fairy Tail Zero.
Adding on to what other people that say about "Fairy Tail" means "endless adventures and mysteries", I also believe Mavis also meant the name was based of on Zera.
In Fairy Tail Zero, Mavis started to form a guild at the place (Magnolia) where Zera's dad and the Red Lizards guild have all been executed. Mavis wanted to help the village to fulfill Zera's dream, by starting a new guild with a name related to her. There was a scene where Mavis pointed out about Zera's hair looking like two tails and that Zera said that fairies would always be in Mavis' heart and where did Zera go? Right back into her heart, referring Zera as the fairy with tails.
Now that you might have thought about it, Zera might actually been the fairy Mavis has always been looking for.
My personal theory is that it represents Mavis's search for the faeries with Zera.
In one scene of Fairy Tail Zero, Zera and Mavis are talking, and Mavis points out that Zera's hair looks like tails. To also explain another idea about the faeries, when Zera "disappeared" and they hugged in the pond, it looked like there's a little ball of light -- a fairy. And when Zera disappeared and Mavis was crying, it looked like Zera turned into a group of faeries. When Warrod, Yuri, and Precht heard her voice, it looked like a faerie was flying past them.
