Dye armor

[Kirika-]

Complete the quest:
https://ffxiv.gamerescape.com/wiki/Color_Your_World
to unlock the ability to dye weapons and gear.

NOTE: Not all weapons or gear can be dyed.
If a gear piece or weapon can not be dyed, it will both have the description text of "Dyeable: No" in it's item window and no option to dye via the right click menu.

Normally, if you see someone with a gear/weapon that is dyed, but it seems the piece isn't dyeable, it is likely they are using a gear/weapon that has the same base appearance as the non-dyeable version, of which can be dyed.
So, look at the item's description window to make sure it's name.

[Rongway]

[general dye info]

I think they're specifically asking about the job-specific artifact gear.

The ARR AF1 gear (Lv45/50) comes in three versions: the IL50/55 gear, the IL90 AF1+ recolor, and a third version that is dyable. You have to have an Alchemist craft the appropriate augmentation kit and then take that along with the original gear to Rowena's castle in Mor Dhona to trade for the dyable version.

The HW AF2 gear should be dyable as soon as you get it from the vendor since they only sell the upgraded dyable version now.

SB AF3 requires doing Eureka.

ShB AF4 requires doing a certain trial extreme (the one that doesn't drop a mount):

Memoria Miseria

EW AF5 requires completing the related role quest line.