The name Hudson's River appears on a number of English maps published in the seventeenth century, but on very few Dutch maps. The best source for information on this is Philip Burden's Mapping of North America (1996). Stokes, Iconography of Manhattan is also useful. One example of an early map with the name "Hudsons R." is Henry Brigg's "The North Part of America" (1625?). Burden and Stokes argue that the appearance of this name on an earlier map published in Amsterdam by Goos indicates a common (English) source. I believe Burden mentions that there was an earlier English map which used Hudson's name to designate the river. On the general subject of Dutch and English cartographic rivalry see Benjamin Schmidt, "Mapping an Empire: Cartographic and Colonial Rivalry in Seventeenth-Century Dutch and English North America," William and Mary Quarterly 54 (1997), 549-78.

Hudson was, of course, English, but he sailed to "his" river under the Dutch flag. There is a whole body of literature dealing with such subjects as whether Hudson was a traitor, or if his discovery should be considered Dutch or English. As good an introduction as any is Donald Johnson's recent Charting the Sea of Darkness: The Four Voyages of Henry Hudson. I think this whole discussion is beside the point. In my view Hudson was basically an opportunist who wanted to discover a route to China, and was not concerned about the implications of his voyage for future Dutch and English claim to North America. What matters is the uses that Dutch and English propagandists made of his "discovery" to back up their claims.