The idea of submitting a dispute to be resolved a third person seems to be as old as the civilization itself. No wonder then that search for true sources of "first arbitration" appears to be in vain. This however, shall not prevent us from trying to acquire at least SOME general knowledge about the documented sources of lex arbitri. The oldests traces of laws governing arbitration that I have come across come from the beginnings of the Roman Empire.
The word "arbitration" itself has latin origins and comes from the word "ar" which means "to" and "bitere". Adding one and one together will lead us up to the conclusion that "ar bitere" means "to go". Arbitrator was then supposed to be a person "going to" somewhere. But.. where to exactly?
In one of the ancient texts written by Cicero, he "praises one of the regulations coming from the The Law of the Twelve Tables for its moderate sanctions. The articles expressed therein stated that one must not kill a thief during the day, except when he (the thief, that is!) carries a dangerous tool. Then, whenever the thief means to use that tool, one may kill him but beforehand one shall shout (literal translation in my possession actually states: "make some noise"), so people, particularly neighbours, could hear and come to see the happening." The people, "those who come" - called arbitris - were summoned so they could state later on that the deadly blow was administered in accordance with the law. Everything performed lege artis, however I believe that for the now deceased thief the legality issue was of a rather disputable value.
Another step in Roman evolution of arbitration, much closer in its nature to adjudication, was pignus. Pignus was basically a bet. The parties to a dispute bet on who was right and who was not. Each of the parties in conflict had to give an item of value to a third person and then the party who was said to be right on the subject of the dispute by that third person won the bet.
Decades later, a so-called compromissum is born. Compromissum was very much like an arbitration agreement/clause we now nowadays - it could contain very specific instructions governing the procedural aspects of dispute resolution. It was, however, considered a private agreement without protection of public law. What it meant for the parties was that they could not execute the arbitration agreement with the help of officials. To solve this problem another type of agreement, or rather agreements, called promissiones, had to be made. The parties promissed to each other to pay a specified amount of money if they failed to obey the regulations of the arbitration agreement. The promissiones were protected by the public laws.