Arbiter is a term used by FATE to designate the last person to make contact with the Frozen Flame. The Prometheus Circuit was designed to lock out everyone but the Arbiter once contact was made; it did so after Serge touched the Frozen Flame to heal.
The role of "Arbiter" probably had a larger role lost with the peripheral storyline that did not make it into Chrono Cross (see the last paragraph at Chrono Cross Condensed Plot Summary). The Arbiter is hinted as being a mediator between Humans and Lavos, capable of guiding his species. It is suggested that if the Arbiter does not make a favorable decision, he will become integrated with Lavos, spelling doom for the rest of his kind. Ultimania speculated that Schala was an Arbiter in Zeal, probably controlling the Flame's use in the Mammon Machine, and that this link was what caused Lavos to merge with her and become the Time Devourer. Part of this depends on whether the Frozen Flame's creation was an accident, and whether Lavos normally selects a sort of powerful acolyte (such as Queen Zeal) in the vein of the Arbiter.