I heard that a possible explanation of currency was blood feuds. Basically, if some extreme wrong was done on one family by another, instead of having a big conflict over it, the mediating government would give the wronged family a 'proof of wronging' that was legal tender. The generic value of that proof was either to represent a unit of flesh if the debt was not paid, or an amount of a specific valued-but-common good, but which could be sufficed with other goods in negotiations (some goods that were suggested were a pound of grains or a number of wolf pelts). I don't know how accepted that idea is, but it sure beats the old-and-cliche barter system idea.

Video that introduced me to this hypothesis.