The book Gotham: A History of New York City to 1898 doesn't seem to say much about reactions to the name change itself. It seems to have been a minor point when compared to many other things that were going on at the time. Here's a bunch of background to give context (mostly from Gotham, quotes from the book):

New Amsterdam was basically a company town run by the Dutch West India Company and founded only a few decades before all this. Peter Stuyvesant, appointed Director-General of New Netherland by the company, and the many people of New Amsterdam had clashed over various issues for about twenty years before the English took over.

One issue, for example, was municipal government. When Stuyvesant arrived the town was run by himself, a deputy director, and a schout (something like a bailiff). But the people wanted a proper (Dutch type) municipal government. Things got very heated between Stuyvesant and "the commonality" and for a time it looked like the Dutch States-General might revoke the company's charter and assume direct control. The States-General decided not to make any major changes due to the growing threat of war with England, in the 1650s. But Stuyvesant was told to give New Amsterdam a proper "Burgher Government", with mayors and aldermen and such.

There were years of power struggle between Stuyvesant and the municipal government, or magistrates. Stuyvesant's authority was constantly being challenged. The First Anglo-Dutch War hadn't involved New Amsterdam much, I think, but it weakened the Dutch West India Company. Then there was the Restoration in England and a new phase of trade war between the English and Dutch.

This is when the Duke of York came into the picture, intent on smashing the Dutch West India Company. At the same time there was a growing number of English towns being founded in New Netherland, mostly from New England. Agitation against the Dutch got out of hand on Long Island. As New Netherland "disintegrated" the Duke of York got his brother, King Charles II, to grant him New Netherland. Charles II considered the Dutch colony illegal—or at least he told a Dutch ambassador that they had no right to be there.

York sent a force to secure his new estate, large enough to easily defeat the local forces. Stuyvesant prepared to fight, but had very little to fight with. And "after years of acrimonious strife with Stuyvesant...few of its residents were willing to risk themselves and their property for him or the West India Company." York's man, Colonel Nicolls, guaranteed "every man his Estate, life, and liberty". There would be no explusion of settlers, no seizing of property, no required changes in religion, language, or customs. Stuyvesant made sure no one saw Nicolls' letter, nor another, repeating the offer, sent by Governor Winthrop of Connecticut.

But word leaked out and soon Stuyvesant was faced with a crowd of angry new Amsterdam residents and was forced to read the letter aloud. The next day Stuyvesant received a petition from about one hundred prominent residents, including his own son, demanding capitulation against "so generous a foe". Stuyvesant finally gave up and the formal Articles of Capitulation were drafted up.

The Articles of Capitulation were, as Burrows and Wallace put it in Gotham, "the essence of moderation, conciliation, and even compassion." The Dutch of New Netherland had their freedom to stay or go as they pleased, with all their possessions. Those who stayed had to take an oath to the king, but otherwise could live pretty much as they had. They would not be compelled to take up arms against the Dutch United Provinces in the future. Existing contracts would remain enforced. The Reformed Church could still collect taxes, run schools, hold services in Dutch, and so on. Religion freedom was guaranteed—something Stuyvesant had fought against. Very few Dutch left. The former magistrates even thanked the Duke of York for sending "so gentle, wise, and intelligent a gentleman" as Nicolls, who became governor.

So, that is the larger context in which the name change occurred. Apparently Nicolls announced the name changes almost immediately after the formal capitulation and changing of the flags: Fort Amsterdam became Fort James, Fort Orange became Albany, and both New Netherland and New Amsterdam became New York. But it seems that any resentment over the names was overshadowed by the larger context.

In time there was some discontent and various issues, not least the old one of municipal government. The Dutch regained New York in 1673 during the Third Anglo-Dutch War. The Dutch admiral Evertsen promptly renamed the town New Orange. But within a year it was returned to England in the peace treaty that ended the war, status quo ante bellum.

A final point. In the late 1600s there was a lot of movement of peoples around the American colonies. Towns and demographics changed quickly. By 1680 New York's population had grown by about 3,000 (a lot back then), and about half of the Dutch who had lived there during Stuyvesant's surrender were gone, either dead or moved away. In a period when everything was in such flux name changes were mostly just another thing subject to change.