Ah, just something to note, there is no good, clear indication in CC that the "Frozen Flame's Arbiter" is really anything more than a security clearance at Chronopolis. Serge was an Arbiter, but the game implies that there were others before him (indeed, Belthasar may also be an Arbiter; which is to say, Belthasar might have the security clearance necessary to access the Frozen Flame).
Thus, Serge became the Arbiter because he was the one who touched the Frozen Flame. In doing so, the Prometheus circuit activated, gave him an internal security clearance level of "Arbiter," and locked everyone else out (specifically, he locked out anyone without Arbiter Level clearance... which was everyone, since that security clearance hadn't been granted in an age).
It is very important to note that Serge's status as an "Arbiter" is granted by and recognized by Chronopolis. One could say that Robo chooses Serge, not the Frozen Flame.
However, it should also be noted that there are two different "Arbiter" status' thrown around. The one used throughout most of the game seems to be in relation to the above; Serge is Arbiter of the Frozen Flame as determined by Chronopolis. But then, at the end of the game on Opassa Beach, the Ghost Children say that Serge has been chosen by the FF as an arbiter to judge the future form of the TD; Serge is the "Arbiter" as determined by the Frozen Flame itself.
This also reveals to us a little about the character of the Frozen Flame. It is not the Entity, obviously, but it also isn't the Time Devourer (why would the TD give a human the power to determine its own fate?). It might be Schala, as it wants to heal Serge and have him judge the TD, but the Frozen Flame is connected in some manner to Lavos as well. Thus, the TD might actually be a three-part entity. The Body of Lavos, the Soul of Schala, but the Spirit of something else (the Time Devourer). Indeed, the TD goals seem to be rather detrimental to Lavos, as Lavos is a parasite focused on the preservation of its kind (the TD would threaten its entire race). Thus, the Frozen Flame might represent both Schala and Lavos, as a sort of consciousness of the TD.
Under such a circumstance, Serge seems to have been chosen for a simple reason: he was available.