First of all you need the power and reputation to protect the village against most if not all threats. This is like how they choose Tsunade for her strength and her place as a Legendary Sannin. She might not be the strongest in the Village (Jiraiya and perhaps Danzo is above her), but she is strong enough to make people wary of attacking the Leaf. Second you need the experience to lead other people, know how to make hard decisions and be able to lead a organization like a Hidden village. Third is that you need the charisma to make people follow you and your orders, even if they don't understand them. Then there is a fourth, although it is more like a tradition, to be family with a former kage of the same village, like how Tsunade is affiliated with three of the former Hokages and Naruto with six former Hokage.