As Jaap Jacobs writes in The Colony of New Netherland: A Dutch Settlement in Seventeenth Century America:

"Owing to lack of data, the exact chain of events that led to the founding of New Amsterdam is uncertain."

That's in large part because none of the documents written before 1638 in the Dutch colony survive. It's not known exactly what happened, but Russell Shorto speculates in The Island At the Center of the World that after disasters by some of the early administrators of the colony led to their dismissal, these "early governors...likely took the records of their administration back to Amsterdam with them to aid in their defense." And then they were not preserved. In any case, once the old Dutch manuscripts began to be collected by the state of New York in the 19th Century, they found that nothing before 1638 survived. All information before that date comes from mentions in later documents, and from the few that were written in Amsterdam that mention events going on in their New World colony. One of the biggest sources of information is by a writer named Nicolaes Janszoon van Wassenaer, who published Historical Narrative of all the Most Memorable Occurrences Which Have Come To Pass in Europe in 21 parts during the 1620s, and a few sections of a few of the parts recount some of the goings-on in New Netherland at the time. Pieced together, the early history of the colony goes something like this:

The first settlement by the Dutch in their "New Netherland" colony was on an island in the Hudson River near present-day Albany. This was established in 1614. It was no more than a small military fort occupied by about a dozen soldiers at a time. It was also used as a trading post for the Dutch to trade goods with nearby Indian nations in exchange for furs. The spring thaw of 1618 flooded the fort, and instead of rebuilding, it was abandoned.

In 1621, the Dutch West India Company was established and began to make plans to settle the North American continent on a more permanent basis. The first ships over arrived in 1624 with probably no more than a couple dozen settlers intending to stay permanently, along with some military personnel who would be swapped out with other military personnel on a regular basis.

Dutch law had it that a land claim was only valid if there were inhabitants living on the claim; this was different from English law, which said that a claimant just had to set foot on it before any other Christian nation did, and the claim was valid even if they subsequently left. So, for the Dutch, who wanted to lay claim to the Hudson River, the Connecticut River, and the Delaware River, when they arrived in 1624, they split up the few permanent settlers along with a small military guard into four separate settlements. One was again on the same island in the Hudson near Albany. Another was on Governor's Island in New York Bay, not far from Manhattan. And then another on the Connecticut River, and another down in Delaware.

About nine months later, the first governor of the colony arrived, Willem Verhulst, who went to the settlement near Albany. By then, he was told that all the settlements were in trouble and on the verge of failing in the face of discouraged colonists. He sent word back to Amsterdam, and in April 1625, the Dutch West India Company sent instructions to him to pick one of the settlements and then consolidate all the colonists at that location.

Whether he did this or not is unknown, but if he did, he probably picked the colony at Albany. But either way, the colony at Albany failed shortly after that. Verhulst had been instructed to trade fairly with surrounding Indian nations, and not get involved in their politics or wars. Some of the colonists seemed to be upset with Verhulst, though, because he was ripping off some of the Indian nations, and being fairer to others, and it was causing tension. Most catastrophic, though, was that he entered into an alliance with the Mahican nation, who were engaged in an on-again off-again war with the neighboring Mohawk nation. When the Mahican called on their Dutch allies to help fight a war in early 1626, Verhulst obliged and ordered some of the men to help out. And while gone, the Mohawk attacked the Dutch settlement and killed many.

The surviving colonists then held a vote and banished Verhulst back to Amsterdam, while at the same time electing Pieter Minuit as the new leader of the colony. Probably immediately, he ordered the colony at Albany abandoned, and the Delaware and Connecticut colonies abandoned, too, and gathered them all at the colony on Governor's Island in New York Bay, at the mouth of the Hudson.

The settlement on Governor's Island did not last long. It's a small island, and had no freshwater. Minuit then began negotiations with one of the area Indian nations to purchase Manhattan island. Notably, there was more than one Indian nation that occupied the island, but all Minuit wanted at first was the southern tip, though there has since been some doubt that the Indian nation he negotiated with actually had any claim to that land.

Regardless, the Dutch then moved from Governor's Island to Manhattan island, and it remained the center of the Dutch colony for the remainder of the Dutch period, and the English did not change the location when they captured the colony. What was advantageous about Manhattan over any other location? As Shorto writes in The Island At the Center of the World:

"It was large enough to support a population, small enough that a fort located on its southernmost tip could be defended. Its forests were rich in game; it had flatlands that could be farmed and freshwater streams. It was situated at the mouth of the river to which Indian fur-traders came from hundreds of miles around, and which connected to other waterways that penetrated deep into the interior. It was also at the entrance to the bay, located in a wide and inviting harbor that seemed not to freeze over in winter. It was, in short, a natural fulcrum between the densely civilized continent of Europe and the tantalizingly wild continent of North America. It was the perfect island."

It should be noted that Shorto's book leans a bit more into the realm of pop history rather than the scholarly, however, it is well-sourced, even if it's end-noted instead of foot-noted. And this prosaic passage is based on the events that happened both before Minuit made the purchase, and events that happened after.

And once chosen, as historians Mike Wallace and Edward Burrows write in Gotham: A History of New York City To 1898, it was in the interests of the Dutch West India Company to keep the major settlement there. One of the main purposes of the Company was to make money in fur-trading with Indian nations. Keeping the colonists centrally located allowed the Company to run the colony in part as a fur-processing factory, and it also cut down on overhead, by requiring less military personnel. With the population being so low for the first couple decades, spreading the colonists out as had been the original plan would have imperiled the Dutch West India Company's financial mission to its shareholders:

"Settlers would require constant support and protection—both of which cost money—and the more there were, the trickier it would be for the Company to maintain its authority. Besides, settlers would inevitably squabble with the Indians over land and livestock, jeopardizing the flow of furs into the company’s storehouses."

Thus, the Dutch West India Company focused its efforts in Manhattan and the surrounding areas. It also cost money to purchase additional land from Indian nations, so the Company only did this later on, once it came to be in their financial interests. And the land they bought at first was almost entirely in the area surrounding Manhattan.

Once the colony was relatively successful, there was no immediate reason the main settlement couldn't have been moved to Long Island, or northern New Jersey, or the Bronx, or Staten Island. In fact, the Dutch made more purchases and farmers did begin colonizing these areas. But due to Indian attacks, and competing land claims by the English, Manhattan remained the most sensible center of the colony. What is now Suffolk County, Long Island, was occupied and settled by the English, so trying to move New Amsterdam to the Bronx or Queens would have opened up the Dutch to a less-defensible attack from the English.

And settlements in northern New Jersey and Staten Island both failed. Another of the early failed Dutch governors was named Willem Kieft, whose rocky tenure came to an end after Kieft's War in 1645. He had demanded some of the Indian nations to pay monetary tribute to the Dutch, among other actions that caused turmoil. Kieft's War was the end result, and the settlement on New Jersey was massacred. The isolated farmers on Long Island and Staten Island were attacked, too, and these settlements were also abandoned.

It wasn't until later that the Dutch attempted to re-establish settlements on Long Island and Staten Island, but they were always vulnerable to attack. The Peach War in 1655 and the Esopus Wars between 1659 and 1663 saw further attacks on these areas, to the point that the new Dutch governor, Pieter Stuyvesant, ordered the isolated farmers on these islands to abandon their farmhouses and form fenced-in villages, which they mostly did.

(cont'd...)