I couldn't wait and took a read on all of the BLU quests in Garland tools website. It's actually pretty good! Well written and with relatable characters with some funny moments. I was greatly interested on how someone could feasibly roleplay a blue mage and could see some point to support that. Spoilers below.
1) Martyn is the main NPC. He used to be an arcanist that got sent to the New World and learned about blue magic there, but his funds got cut out because the Arcanist Guild thought that was a worthless quest. Although travelling there is not something common, it's possible to learn blue magic with the Whalaqee people.
"The name's Martyn, and today I present to you a wonder of the New World. I call it “blue magic”! Beyond the Indigo Deep live the Whalaqee, who have their own unique brand of spellcastin'." - Martyn
2) He sells blue mage crystals and spellbooks, with instructions on how to learn spells from monsters, but people claim it's a scam. In reality, according to Martyn, the customers are just lazy and won't take time to read the instructions on the spellbook:
"I'll have you know I poured my heart and soul into those crystals, and the spellbooks explain how to learn the magic. It ain't my fault if the buyers can't be arsed to read them." - Martyn
The yellowjacket that investigates him ends up convinced that it's not in fact a scam, but she claims that the way Martyn proposes his customers to learn blue magic is too irresponsible because it exposes one to death hazards, so she ends prohibiting him to keep his business:
"Not so fast, you blue-haired twit! I'll admit, the spell worked as you said it would. But you can't sell those soul crystals without adequately explaining the risks! It's irresponsible! Some might even call it deliberately neglectful.
You're not under arrest, but from this day forth, you are strictly forbidden from peddling any more of your wares. If I hear you've sold so much as a blue handkerchief, I will hunt you down and throw you behind bars!" - Yellowjacket
3) By the end of the questline, Martyn say that he'll save gil to open a Blue Mage's guild, so he can spread the knowledge in Eorzea. Possibly with more... safe ways now.
"In fact, I'm thinkin' about openin' a Blue Mages' Guild, partly as a way of rubbin' my old guildmates' noses in it for rejectin' my research, but also to help spread the knowledge across Eorzea. I'll need a fair bit of gil to get the idea off the ground, though." - Martyn
So, we learn that the WoL canonically acquires a BLU crystal to prove to the yellowjacket that Martyn is not a scammer, but for the regular roleplayer, I suppose going after one, could mean the following:
- You learned in the New World as well, and acquired or crafted your crystal there and knowledge on how to learn from monsters. This would be a bit uncommon route, but totally possible.
- You were one of the purchasers of Martyn's blue mage "kit" (crystal, spellbook and blue attire). Apparently it's a popular item, to be able to grab Limsa's authorities' attention. This appears to be the best route yet.
- You could be one of the future enrollers of his Blue Mage guild, but I suppose that's is a plot line that will happen when the next tier of 10 levels are unlocked with new quests.
Interesting facts:
- BLU's crystal role is basically a vessel to store monster knowledge, pretty much like MCH's crystal: Unlike other soul crystals, the surface of this multi-aspected stone has yet to be carved with the record of past deeds. - BLU crystal tooltip
- Blue magic draws aether from the environment
- The whole Blue Mage dapper aesthetic and the carnivale that we get to see in the previews, the mask theme and such, it's not a "tradition", but something that is made up during the storyline. Probably a "traditional" blue mage wears the garnments of a Whalaqee. We get to meet a NPC from there during the quests and he wears the New World outfit, sold in the MB. The outfit that Martyn originally wears and the one he sells with his blue mage's kit is indeed blue, but it's more like a regular adventuring outfit. The model offered to the player is called True Blue and is the same that Ramza wears in Ivalice raid arc.