Plus they just sound magic: tourmaline; amethyst; plagioclase feldspar. 
Besides, what else is crystallised magic supposed to look like? 
More seriously, crystals have been seen as magical for millennia, with accounts going back to the Sumerians and Ancient Egyptians. (Heck, the word amethyst comes from the Ancient Greek for “not drunk” as it was believed to cure hangovers.) It’s not just something that SquareSoft made up for Final Fantasy; it’s got a long mythological history behind it. And if you’re writing a story where there are a number of magical objects that can be broken one by one to raise the stakes and you’re writing in a setting where all myths and legends are real to an extent (definitely not talking about anything specific here, no), then having a set of crystals (say, corresponding to the four classical elements) is something that makes a lot of sense.