A grietenij is a municipality. The term is mainly used in Friesland. In 1851, it was changed to gemeente, the common Dutch word for municipality. In parts of Groningen, the term grietenij was also used to refer to the local courts.
A grietenij was led by the grietman, whose function was similar to that of a mayor.
I have an ancestor William Gritman. The first record of him is 1685, baptizing his son Wilhelmus at the Dutch church in Flatbush (Brooklyn), New York. This was after the conquest of New Netherland by the English. It is unknown whether he was the original immigrant, or was born in New Netherland.
His son Wilhelmus baptized several children at the Dutch church in Jamaica, Long Island, starting in 1702. He is listed as a subscriber to the church in 1715. The original Jamaica church records survive, and the recorder (probably the dominie) spelled the name variously as “Grotman” or “Gretman”.
After the family assimilated the name has been spelled “Gritman”.
Given that the Dutch did not use surnames until much later, I assume the family had to choose a surname in New York.
You point out that “Grietman” was a term primarily used in Friesland. I’m interested whether you think the selection of this surname could indicate that William’s ancestry was Frisian?