Back to this!
Coin showed up in the fourteenth century, but back then it only meant a “wedge-shaped piece used for some purpose”. See, the thing that stamped metal—such as coins—was wedge-shaped, and so in the fourteenth century it evolved to mean something stamped or a stamped piece of metal made into money. That usage was influenced by Old French, which is also the language coin comes from, although there it’s coing. It comes from the classical Latin cuneus, which means wedge (also it can mean “group”, but that’s not really relevant here). Fun fact, there is also a word “quoin” which I hadn’t heard of before but means a cornerstone or the corner of a wall—generally something wedge-shaped. It’s literally just a variant spelling of coin that stuck close to its original meaning.
Currency is relatively recent, having shown up in the mid seventeenth century meaning… a condition of flowing. Like the current of a river. Yeah, this is another weird stretch of the word. Currency meant a “state of fact flowing from person to person”, you follow? Which evolved to a sense of “continuity in public knowledge” and then the current medium or exchange of money in the eighteenth century. Basically, anything that was currently money was currency. The word comes from the classical Latin currens, current, and its verb form currere, to run, which is descended from the Proto Indo European word kers-, to run. And that’s definitely a word we’ll have to look into sometime.
Capital in terms of money showed up in the early seventeenth century, from the Medieval Latin capitale, stock or property, and classical Latin capitalis, capital, chief, or first. Every other version of capital comes from there, too, although this isn’t too surprising since they all refer to something that’s foremost, whether a capital letter or a capital city. Hell, even capital, the head of a column or a pillar, is related because it’s the head of the column or pillar. I guess people thought of their wealth and property as being the principal thing in their lives. Yeah, I can see it.
Speaking of wealth, it showed up in the mid thirteenth century, where in addition to meaning “prosperity in abundance of possessions or riches”, it meant happiness, another thing I can understand all too well. The word is actually from the Middle English wele, well being, which actually spawned another word I’ve never heard of: weal. That word is wela in Old English, meaning prosperity, from the West Germanic welon-, from the Proto Indo European wel-, to wish or to will. Wealth is related to both will and well, because when you’re doing well and have the things you wish, you’re wealthy.
Tony Jebson’s page on the Origins of Old English