Coins only make sense in an economy that has taxes, and they originally came about as a by-product of warfare. Perhaps coins would be proffered when sending trade caravans to other civilizations, as an excise tax, say 5% of the total value of your caravan has to be provided in coin to the civilization or the caravan would be turned back at the border. Likewise, any incoming caravans could be excise-taxed on entry into your civilization's territory, and whatever rate you set the tax too would determine how much goods they would send: a 50% excise tax would result in a caravan carrying less in trade goods, say.
The other effect would be to pay dwarves for their military services in coin depending on how many days out of the year they are active. The soldiers could then contract services from civilian dwarves, such as paying a craftsdwarf to decorate a cabinet of theirs with bones or something. Dwarves could be free to create items for their own consumption and use, but any such unsolicited items would require a workshop tax on production, so that anything a dwarf produces on their own would result in some coin going back to the government. In this way some things could take care of themselves, like if a soldier wants a leather waterskin made out of their favourite animal's hide, and have it decorated, he could issue the contract to any dwarf who has the requisite skills to do it, and the soldier would return his current waterskin to the stockpile and keep the new one on him as an "attached" item, similar to how dwarves already grow attached to certain items. Soldiers could also contract their favourite foods as their rations in the above process, and return their current rations to the pile. This would make for more happy dwarves and more needs being met independently of micro- or macro-management.
For added fun, dwarves may sometimes cast coins into wells that they frequent, generating happy thoughts depending on the value of the well, and extremely happy thoughts if somehow their wish should come true. On the flip side, other dwarves that have made wishes at the same well may become annoyed by the fact that their own wishes hadn't come true, generating bad thoughts. Dwarves might start fist-fights with thieves that try to dive and take any such placed coins, even if ordered to by fortress management, meaning any such deposited coins would have to be forbidden. However, the coins could be declared as charity for worthy causes, like helping orphaned dwarf children, or provided medical attention to injured pets and such, in which case the well-wishers might not be so perturbed.
For even more fun, some merchants might try to use debased coins and pass them off as more valuable coins, and so could the dwarves against other civilizations and even on each other, generating many unhappy thoughts for aggrieved parties if they should find out, and diplomatic breakdowns between traders and civilizations. Likewise, soldiers paid in debased coins might grow to resent that and generated more unhappy thoughts.