The Hudson River and Hudson Bay are named after two different Hudsons.
Both are named after Sir Henry Hudson, an English sea explorer and navigator in the early 17th century. He made two attempts on behalf of English merchants to find a prospective Northwest Passage to Cathay (today's China) via a route above the Arctic Circle. Hudson explored the region around modern New York metropolitan area while looking for a western route to Asia while in the employment of the Dutch East India Company. He explored the river which eventually was named for him, and thereby laid the foundation for Dutch colonization of the region. Hudson discovered a strait and immense bay on his final expedition while searching for the Northwest Passage. In 1611, after wintering on the shore of James Bay, Hudson wanted to press on to the west, but most of his crew mutinied. The mutineers cast Hudson, his son and 7 others adrift; the Hudsons, and those cast off at their side, were never seen again.