3 ideas
Davies describes banking, credit records and units of account in Mesopotamia long before coinage appeared, with temple and palace accounts kept in grain and silver weight units. Coinage came as a later technology within an existing monetary practice, and pre-coin accounting was one of several roots of money rather than its single origin.
The form money takes in a society is driven heavily by political, religious, military and social demands such as tribute, fines, bride-price, sacrifice and state finance, not only by the convenience of trade. Looking at a monetary system through who needed to pay what to whom, and why, explains its shape better than a pure barter-to-exchange story.
Around the seventh century BC in Lydia, lumps of electrum began to be stamped with official marks certifying weight and purity, so they could be accepted by count without weighing and assaying at each exchange. This built on earlier practice such as guaranteed Cappadocian silver ingots rather than inventing state-guaranteed money outright. Coinage then spread through the Greek world.