Horn

Eiko has a horn on her forehead as she's not fully human--she's a member of the Summoner tribe, a race of people who, living in Madain Sari, built a wall dedicated to the Eidolons, prayed to them, and communicated with them.

Eiko uses her horn both to communicate with Eidolons and to sense their thoughts and feelings. One can assume that the horn is an important component in this: Dagger, the only other living Summoner, has lost her horn, and this sixth sense as well.

During trance, Eiko's horn grows longer, possibly an indication that when a Summoner grows more powerful their horn will grow as well. However, since she's the only Summoner in the game with a horn, this can't be proven.

The Summoner's horn didn't originate in Final Fantasy IX, though the horn actually being a part of the body might have. The traditional Summoner's garb found in Final Fantasies V, Tactics, and most recently XI Online, all include a horn in their headgear.