After getting your first Core Crystal, Gramps mentions that with a high enough luck stat, you could spawn a special Blade. What kind of luck threshold is there and what Blades can you unlock this way?
It would appear that special Blades are rare, "named" Blades that can be gotten from any core crystal. Some specific crystals give these Blades, but they can also be gotten from common and rare core crystals - rare ones have a higher base chance to give a "special" Blade.
Special Blades have their own unique intro and graphic. Some unlock specific merc missions (Azami comes to mind) where they are at the forefront.
A listing of every unique Blade (story and special) can be found here - spoilers beware.