One of the steps in studying Ten Eyck, is to understand that Ten Eyck is today's common method of writing a name that means at, by, or beside the Oak.
Two precursors that lead up to that. One is that there were other name writing and spellings of the same pedigree from earlier times. Secondly, the subject comes up of the oak. What oak? Story has it, the baby, the first Ten Eyck, was left by an oak tree, and when picked up, was thusly named.
In 2013, I took screenshots from the online Google Books, hopefully the author does not become too angry, as the page was available for free through them, sharing just one page of it for now:
What can be learned from all of the above on this page, is that Ten Eyck is an old name, with spelling variations over the years, and, just because the famous Coenraedt Ten Eyck credited as the forebear of American Ten Eyck, who came to America around the time if not along with Peter Stuyvesant in simplified history summary, arrived from the Netherlands, does not mean that the Ten Eyck ancestry is limited to the Netherlands.
A series of webpages and subwebpages in this site have gotten underway, in the effort to piece together perhaps a less well known branch of pedigree, who had the name Ten Eyck, going back from about the 1920s. See Research Project: The Pedigree of Mrs Gretchen Ten Eyck Thornton.