The hero known for his cunning and strategic mind during the Trojan War is Odysseus. Renowned for his intellect and resourcefulness, he devised the cunning plan of the Trojan Horse, which ultimately led to the fall of Troy. His qualities exemplify the value of wisdom over brute strength in Greek culture and underscore the importance of strategy in warfare. Odysseus's journey, as depicted in Homer's "Odyssey," further demonstrates his cleverness and determination in overcoming numerous challenges on his way home.