The tl;dr is that the commonly told origin of money, right back to Adam Smith, is a-historical assumption that is contradicted by archeological evidence.
The a-historical account is that money was developed to make barter more efficient. This is not what we see in the records from Sumer, and there is similar evidence in the other locations where we know writing arose. Before the neolithic revolution societies economic structure was largely based on social ties and a sense of reciprocal behavior: a sort of informal communism. As sedentary agriculture lead to the development of temple complexes and cities, debt and credit records replaced these. Money was not developed until nearly 2k years later. Barter was actually uncommon in the ancient world and was typically reserved for interactions between strangers such as traders that had traveled long distance and had no social ties to one another. Coinage in particular typically arose as part of a complex that involved raising armies and funding them via taxation.
The commodity theory of money is a-historical, and not what "original money" was.
It's a great book, though Graeber has little restraint in taking pot shots at economists, which might provoke some readers to defensiveness.
So you're saying the first money was essentially "social credit"? I don't disagree with the spirit of the argument, but your good-will credit in your community doesn't have the properties we associate with currency such as fungibility and divisability.
If they used a formalized series of IOUs that allowed for the long term planning of capital then I'd be more inclined to call it currency. But sounds like an interesting book, may have to give it a read.