For me the solution is rather simple and can easily be implemented with current game mechanics.

First of all concerning weapons, all top weapons from each tier, Relic/Artifact/Raid will have a special materia slot attached to them in which you will be able to add a selection of weapon specific materia that add special abilities. For example some special weapon abilities i can think about are, dmg procs, status effect procs, utility procs. For example you can have a Paladin have a dmg proc on his weapon (Holy Strike with 5% chance that does 2-3k dmg) etc, or a Bard that has Paralyze effect on his bow with 5% chance and for utility you can lower the cd of certain skills for each class. Special Materia Augmented weapons will have a specific glow depending on the materia used.

Secondly about armor, you can utilize a similar system as weapons for top tier armor sets, you can even go as far as add materia set bonus combinations. For example if u slot specific materia to each armor piece you will unlock some hidden bonus.

Lastly about the acquisition of the special materia, what you can do is have them be end-game oriented but not make them impossible for casuals to get, otherwise the system won't be used by anyone but a small minority. Token exchange system would work wonders here for excess Law/Esos.

All in all i believe if a system similar to this would be implemented it will add some kind of endgame customization (that is not just cosmetic) to at least differentiate players slightly from each other.