As indicated by the map above, it can be found in Mare Lamentorum (a.k.a. the moon) starting at 8:00 a.m. and 8:00 p.m. It remains available for two in-game “hours” until 10:00 a.m. and 10:00 p.m. Eorzea Time, respectively.
Besides its time-limited availability, this material also has a required Perception stat of 3,600 to gather. Meaning any Miner player must have at least 3,600 Perception to even attempt gathering Ilmenite. This can be achieved via a mix of gear, Materia, and temporary in-game food buffs.
Ilmenite will always appear inside of a single Level 90 “Legendary” node when it spawns. This node will then alert Miners within Mare Lamentorum at the time to its presence with an onscreen message and a gathering icon on the mini-map.
You can use five pieces of Ilmenite, alongside two pieces of Vanadinite, to craft one Ilmenite Ingot. This is a vital component in crafting several Level 90 master recipe items (i.e. high-tier Level 90 gear).
There is currently no other way to reliably acquire Ilmenite in FFXIV. However, players can freely trade it amongst themselves and sell or buy it from the in-game Market Board.