Free banking as a policy ideal is the result of applying the norms of classical liberalism to money and banking. Classical liberalism upholds individual liberty and private property rights, including freedom of contract, under the rule of law. It opposes rule by unconstrained authorities.
Skipping over ancient thinkers, early modern expressions of classical liberal thought on money may be found in the writings of Nicholas Oresme and other Scholastics who denounced the sovereign's debasement of the coinage as a dishonest and tyrannical, a violation of the sanctity of contract that a just sovereign must respect. David Hume in the 1750s refuted the Mercantilist fear that unless the sovereign interfered with freedom of trade and payments the nation's economy would retain too little silver and gold. Adam Smith in 1776, and later defenders of free banking in Britain, on the Continent, and in the Americas (my favorite being William Leggett), applied free-trade doctrines to banking and bank-issued currency. Thomas Hodgskin and Herbert Spencer even dared to defend private coinage. Walter Bagehot defended the principles of free banking, though he thought it a lost cause politically. In the twentieth century the Austrian economist Ludwig von Mises gave new subtlety and rigor to the case for free banking. Friedrich Hayek made somewhat ambivalent cases for gold and free banking earlier in his career, but in the 1970s forcefully called for Choice in Currency and The Denationalisation of Money.
Many leading classical liberals over the centuries, however, have failed to apply their free-trade principles consistently to money. David Ricardo favored nationalization of coinage and banknote issue, and the forced substitution of redeemable paper notes for coins in all but the largest payments. John Cobden, the free trader, supported the nationalization of banknotes. After the Second World War, most of the German Ordoliberals and the Monetarists, led respectively by Walter Eucken and Milton Friedman, made peace with central banking and fiat money. (Although Friedman, it should be noted, reconsidered his position in the 1980s and began favoring free banking and the abolition of the central bank .) The otherwise radically free-market theorist Murray Rothbard favored a 100% reserve requirement, with no allowance for capitalist acts among mutually consenting adults who want to contract around that rule. Alternatives to fiat money, central banking, and deposit insurance were not on the program of the Mont Pelerin Society, a leading international society of classical liberal intellectuals, at its special meeting to discuss the financial crisis in 2009.
A century ago, before the First World War, economic classical liberals almost unanimously supported the ideals of the international gold standard. The "classical" period of the gold standard is usually dated from the United States' resumption of gold payments in 1879 until the suspensions of payments in the First World War. The gold standard was fatally wounded in the First World War and never fully recovered. In practice it had not been a completely market-regulated system even before the war. National governments had long banned market competition in coining by giving themselves a monopoly in the minting of coins. Many similarly banned market competition in the issue of gold-backed banknotes and gave a monopoly to a government-sponsored central bank. National central banks violated the "rules of the game" by interfering with, overriding, or suspending the automatic operation of international gold flows. The centralization of gold reserves, a characteristic feature of central banking, altered the operation of the international gold standard for the worse.
Nonetheless the classical gold standard more nearly approached a self-regulating international monetary system than anything that has followed. Monetary nationalism and monetary statism since the First World War has brought us to our present system of national fiat monies, central banking, and extensive government interference in financial markets everywhere in the world (with the exception of offshore banking havens). The global financial crisis that begain in 2007 has exposed the weakness and non-self-regulating character of the present system for all to see.
Our challenges for the present day, and for this blog, are to discover how we might reform the system in manner consistent with the ideals of freedom. How might we restore healthy self-regulation to our local and international monetary and banking systems?
Here are some references for the above history, which may also begin to answer Brad Jansen's request for recommendations of classic readings.
Nicholas Oresme, De Moneta (Of Currency) [c. 1355], translated by Charles Johnson, in Lawrence H. White, ed., The History of Gold and Silver (London: Pickering and Chatto, 2000), vol. 1; David Hume, "Of the Balance of Trade," in Essays, Moral, Political, and Literary, ed. Eugene F. Miller (Indianapolis: Liberty Fund, 1987); Adam Smith, An Inquiry into the Nature and Causes of the Wealth of Nations, ed. R. H. Campbell, A. S. Skinner, and W. B. Todd. (Oxford: Oxford University Press, 1976); Vera Smith, The Rationale of Central Banking  (Indianapolis: Liberty Fund, 1990); William Leggett, Democratick Editorials (Indianapolis: Liberty Fund, 1984); Thomas Hodgskin, Popular Political Economy (London: Charles Tait, 1827); Herbert Spencer, "State-Tamperings with Money and Banks," in Essays Scientific, Political and Speculative, vol. 3 (London: Williams and Norgate, 1891); Ludwig von Mises, The Theory of Money and Credit  (Indianapolis: Liberty Fund, 1980); Mises, Human Action, 3rd ed. (Chicago: Henry Regnery, 1966); F. A. Hayek, Choice in Currency (London: Institute of Economic Affairs, 1976); Hayek, Denationalisation of Money, 2nd ed. (London: Institute of Economic Affairs, 1978). David Ricardo, Plan for the Establishment of a National Bank  in The Works and Correspondence of David Ricardo, ed. Piero Sraffa with the Collaboration of M.H. Dobb (Indianapolis: Liberty Fund, 2005), vol. 4, Pamphlets and Papers 1815-1823; Richard Cobden in 1840 Parliamentary testimony cited by Lawrence H. White, Free Banking in Britain, 2nd. ed (London: Institute of Economic Affairs, 1995), p. 84; Walter Eucken, Grundsätze der Wirtschaftspolitik (Tübingen: J. C. B. Mohr, 1952); Milton Friedman, A Program for Monetary Stability (New York: Fordham University Press, 1960); Friedman, "Monetary Policy for the 1980s" in John H. Moore, ed., To Promote Prosperity (Stanford: Hoover Institution Press, 1984); Murray N. Rothbard, "The Case for a 100 Percent Gold Dollar" in Leland Yeager, ed., In Search of a Monetary Constitution (Cambridge, MA: Harvard University Press, 1962).