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The Gyan Prasarak Mandali was set up for the education of adult menfolk during the period of Indian freedom struggle. Among the given options, Dadabhai Naoroji is the correct answer as he was responsible for establishing this society.
Annie Besant, option 1, was a prominent figure in the Indian freedom struggle and was involved in various social and educational reforms, but she did not specifically set up the Gyan Prasarak Mandali.
Bhikaji Cama, option 2, was a leading freedom fighter and a prominent voice for Indian independence. However, there is no evidence to suggest that she played a direct role in establishing the Gyan Prasarak Mandali.
G.K. Gokhale, option 4, was one of the early political leaders during the freedom struggle and a mentor to Mahatma Gandhi. While Gokhale was involved in educational reforms, there is no record of him setting up the Gyan Prasarak Mandali.
In conclusion, the correct answer is option 3, Dadabhai Naoroji, as he was the one who established the Gyan Prasarak Mandali for the education of adult menfolk during the Indian freedom struggle.