As a detective aiming to unravel the mystery of the crossword clue ‘Leeward island where Alexander Hamilton was born,’ my first step would be to research the life of Alexander Hamilton. A quick dive into Hamilton’s biography would reveal that he was born in the Caribbean, in a British colony that was later part of the Federation of Saint Kitts and Nevis. This information narrows down the search to two islands, Saint Kitts and Nevis. The clue ‘Leeward island’ helps further narrow down the search, as the Leeward Islands refer to a group of islands in the eastern Caribbean that are situated towards the North of the Windward Islands. Therefore, with these pieces of information, it becomes clear that the answer to the crossword clue is ‘NEVIS,’ one of the Leeward Islands located in the Caribbean Sea, east of Puerto Rico. By connecting the dots and using insights from Hamilton’s biography and geography, the answer is revealed, concluding the quest to unravel the mysterious crossword puzzle.