Dodson was born in Katherine, in the Northern Territory. He is a member of the Yawuru people, who are the traditional owners of the land around Broome, in Western Australia. His brother is Patrick Dodson, who has become a well-known Aboriginal leader as well. Their parents died when they were still children. When this happened, Mick moved to Monivae College, a boarding school in Hamilton, Victoria. He graduated in law from Monash University in 1974. He was the first indigenous person to graduate from law in Australia.