Why did New York become the financial hub of America, rather than one of the other big cities?
There are several reasons:
New York was a natural trade center for a few key reasons.
Access to Upstate NY
Upstate NY was very important place in commercial history. The settlement of New York was driven by access to beaver and other furs, and the Hudson River was the 16th century equivalent to an interstate highway, leading right to the port of New York City.
Later, as fertile land in New England was accounted for, agriculture became a huge force. The feudal-style Dutch Patroonships produced all sorts of goods, and the Schoharie Valley was called "the breadbasket of the american revolution", and much of the agricultural surplus was shipped down the river to New York. Later, the construction of turnpikes opened up the interior of Central NY, which meant more farming, more credit & banking activity in Albany and NYC and more economic activity.
Most of the English Colonies were formed by relatively homogeneous, religiously-oriented communities. New Amsterdam was totally different -- it was a commercial venture of the Dutch West India Company. Being motivated by money vs. god, there was a diversity of religious backgrounds, languages, etc.
The Erie Canal
The construction of the Erie Canal brought New York to a whole new level -- the flow of goods from Western New York and Ohio all came through New York to be shipped all over the world.
All of this stuff created a scenario where New York City became this dense concentration point or hub for wealth and trade. Until the later half of the 20th century, this proximity made the New York metro region a premier industrial center -- which further strengthened it's role as a financial center.
I would add that Holland was the financial centre of the Earth in 17-18th centuries. NY, having the best specialists in finances simply couldn't fail.
I think the best way to answer this question is to ask: Why did Philadelphia lose its position as the hub of American finance? Philadelphia had first mover's advantage over New York in terms of finance. Unfortunately for Philadelphia, national politics would erase Philly's advantage in the 1830s and 1840s, setting the stage for New York to begin to establish itself as the center of American finance.
Philadelphia's Early Dominance of American Finance
Philadelphia was home to the nation's most important banks before the Civil War: the Bank of Pennsylvania, the Bank of North America, the First Bank of the United States, and the Second Bank of the United States.
Similarly, most of the most important bankers and financiers of that era were Philadelphians (or ended up in Philadelphia): Robert Morris, Haym Salomon, Stephen Girard, and Nicholas Biddle. As I discuss below, Jay Cooke and the Drexels also deserve serious attention.
Jackson, Van Buren, and Philadelphia's Long Slow Fall from Grace
As one New York Jacksonian put it, "The general place of the deposit of the revenues of the government must become the money market of the nation" (Bray Hammond, p. 392). As long as the Second Bank of the United States survived as that "general place of deposit," this meant that Philadelphia would reign as the money market of the nation.
Andrew Jackson's determination to kill the Second Bank for ideological reasons is well known. Jackson received invaluable assistance from his Secretary of State (and later Vice President) Martin Van Buren. Van Buren's plans did not involve creating any replacement for the bank--in part because Van Buren hoped that much of that money would flow to New York's banking system.
During his incredibly short stint as governor of New York, Van Buren had helped to pass the Safety Fund Act of 1829. The Safety Fund is a remarkably influential piece of economic regulation, and its influence is still felt in the modern FDIC. New York's banks were awful prior to this act, and the Safety Fund provided some stability to New York banking until 1843, but even when the Safety Fund faltered it was soon replaced by a well-regarded Free Banking Law. So by the 1840s, New York arguably had the best-regulated banking system in the nation. New York's banking stability was further enhanced by the New York Clearing House, organized in the 1850s.
But the important thing for the Bank War is that the Safety Fund system united the power and interests of all of New York's banks into one "mighty influence" that answered to the leadership of the Democrats in the "Albany Regency." As Erastus Root reported of the New York banks,
Yet the alliance of dozens of New York state banks seemed less threatening to most Americans than one large bank in Philadelphia:
To some extent, bankers in all the states of the Union (other than Pennsylvania) saw that the destruction of the Second Bank of the United States would redound to their financial benefit. They all threw their weight behind Jackson's effort to kill the bank. But only the bankers of New York were well organized enough to step in and take control of this opportunity.
What about Buttonwood and the New York Stock Exchange?
Philadelphia is home to the oldest stock exchange in the US, the Philadelphia Stock Exchange, founded in 1790. The Buttonwood Agreement was signed in New York two years later, but financial markets there were relatively sleepy in their early years. New Yorkers mostly traded old war bonds and Philadelphia-based Bank of the United States stock. In 1817, New York stockbrokers sent delegates to Philadelphia to learn from the original stock. On their return, they founded the NYSE.
So the NYSE is an imitation of the Philadelphia Stock Exchange, and it relied on Philadelphia-based financial institutions for a non-trivial portion of its capitalization. Thus "path dependency" type arguments do not explain New York's current dominance in American finance.
What About the Morgans and Investment Banking?
Modern investment banking also has its origins in Philadelphia. Jay Cooke joined the Philadelphia banking house E.W. Clark & Co. in 1842. In 1861, he founded Jay Cooke & Company, a Philadelphia-based firm that helped underwrite the Union's Civil War effort and innovated many of the most important techniques in modern investment banking.
Even the House of Morgan--the greatest New York financial house in the US in the late 19th century--had important Philadelphia origins. Francis Martin Drexel founded Drexel & Co. in Philadelphia in 1837. Drexel was second only to Cooke in terms of government finance in the 1860s. In 1871, Anthony Drexel approached Junius Morgan about affiliating their houses. The Philadelphia-based Drexel had the money, but the New York-based Morgan had the London connections. As Drexel, Morgan, and Co., the Morgans would go on to dominate American finance for the next half century. (Source: Ron Chernow's House of Morgan)
Respectfully, there is little evidence that Dutch heritage or connections to Amsterdam mattered much to New York's success in finance. I've already argued that New Yorkers in the antebellum era borrowed regulatory structures from non-Dutchmen in Philadelphia. The Dutch character of New York was rather diluted by the 1870s, when the city was beginning to irreversibly eclipse Philadelphia as the American financial center. And during the 1870s, New York finance is largely associated with two non-Dutch ethnic groups: German-Jews and New England Yankees. European Jews and prominent Yankees both had one major advantage in finance: they had overseas ties to bankers in London, Paris, and the rest of Europe. The United States was industrializing so quickly that overseas capital was greatly needed, and so these two groups were able to act as channels for European capital. So New York owes some of its later banking preeminence to its centrality in immigration and thus its many European connections.
Van Buren, of course, was Dutch-American, but his magic was political, not financial.
After the Second Bank
The Second Bank of the United States was liquidated in 1838, and government funds began trickling from Philadelphia to the rest of the country. Much of this money found its way to New York's stable and well-regarded banking system, thus beginning the shift of financial power northward. If Jackson had not killed the bank, Philadelphia likely would have maintained its preeminence longer. (Philadelphia was in the process of improving its connections to domestic markets in the west, which would have gone a long way toward mitigating New York's advantage in trade.) If the Bank had survived, Drexel may never have approached Morgan, or at least the Drexels might have remained the dominant partners in that relationship. German-Jews interested in finance may have immigrated to Philadelphia, not New York City. In the end, however, the Bank did die, and New York had the wealth, the financial infrastructure, and the foreign connections to take over as America's financial hub in the wake of the Bank War.