What Was The Old Name Of New York City?

Depends how far back in time you want to go!

Native Americans didn't have a single name for the entire are that now comprises New York City. The main tribe living in the area were the Lenape, who are part of the larger group of Delaware tribes (linked by language).  The great river flowing thru the city was noted down as the Manahattas River, where the Manahattas people lived (according to the Dutch traders who first colonised the area).  Over time and in English, this got corrupted into "Manhattan".  Modern historians argue that the Manahattas people were really the Lenape, although it's not clear how the Dutch misheard it.

In 1625 the Dutch named their trading outpost in what is today lower Manhattan, "New Amsterdam".  This was part of the larger Dutch colony of New Netherland, which encompassed all of modern New York City plus about 15 miles inland.

By 1665 New Netherland had been annexed into the burgeoning British colonies, although specific assurances were given to the Dutch settlers about local autonomy and tolerance being legislated into their local government.   Many credit this tradition of tolerance in its early history as being a big part of what made New York City into such a vibrant city after that.