Deasy's "Stuart coins" are relics of a sorry period in Ireland's history: its invasion in 1688 by King James II of England. In 1689-90, after securing the allegiance of many of the Irish, James issued a series of new coins that were minted from base metals like copper and brass and debased the existing Irish currency.
This was not the usual invasion. James ascended the English throne on the death of his brother Charles II in 1685, but quickly excited opposition because he was openly Catholic and employed the rhetoric of absolute monarchy, confirming two longstanding suspicions about the Stuarts. When he produced a Catholic heir, powerful Protestant opponents reacted by encouraging his nephew and son-in-law William of Orange to raise an army and invade England from the Low Countries. James fled England, an action which was interpreted as abdication (and euphemized as "The Glorious Revolution"), and William III was installed as King, ruling jointly with his wife, James' daughter Mary.
From France, where he had first taken refuge, James landed an army in Ireland. He was welcomed by the Catholics who had lost land and power as a result of the long Confederate Wars of the 1640s and Oliver Cromwell's conquest in 1649-53. But when William brought his own army to Ireland and defeated James' forces at the Battle of the Boyne (near Drogheda) in 1690, their downfall became total. Catholics who had supported James fled the country, becoming the first "wild geese." The ruling class which eventually became known as the Protestant Ascendancy took shape. And Parliament passed a series of harsh new penal laws in the 1690s and early 1700s to suppress the Catholic population. Edmund Burke remarked that these laws constituted "a machine of wise and elaborate contrivance, as well fitted for the oppression, impoverishment and degradation of a people, and the debasement in them of human nature itself, as ever proceeded from the perverted ingenuity of man."
James' coinage added its own modicum of disaster to this general catastrophe, because it debased the Irish currency by inflating the money supply. Excepting some gold and silver proof strikes, the coins contained no intrinsic value; they are called "gunmoney," because much of the metal in them came from melting down old brass cannons. But, as Gifford observes, they have value as collectors' items: "The coins, though initially as worthless as the Stuart attempt to use Ireland (a bog) as a base from which to retake England, are, of course, rare." Presumably Deasy owns them for their numismatic value, not because they celebrate the Irish Catholic cause. And presumably they exist in this chapter to inspire both Stephen's contempt for the "base" nature of money and his despair about the sorry course of Irish history.
 Thornton quotes from Joyce's "Alphabetical Notebook": "Irish wits follow in the footsteps of King James the Second who struck off base money for Ireland which the hoofs of cattle have trampled into her soil." He notes the relevance of this sentence to Deasy's concern with money and cattle. He does not note that it may also account for the reference to coins being fished from bogs.