"Dude" is an informal term often used to refer to a man or a person in a casual and friendly manner. In the context of "The Big Lebowski," the clue likely refers to the protagonist of the movie, who is known as "The Dude."
Note: The provided meaning of the answer is for the latest New York Times crossword puzzle clue: "My man!". Explanations are generated by AI and may be inaccurate.