Gopal Krishna Gokhale, option 4, introduced a bill in the Imperial Legislative Council in 1911 for the introduction of compulsory and free primary education in India.

Dadabhai Naoroji, option 1, was an influential figure in the Indian National Congress and known for his advocacy for Indian self-rule. However, there is no information or evidence to suggest that he introduced a bill for compulsory and free primary education in India.

Bal Gangadhar Tilak, option 2, was a prominent independence activist and political leader. While he played a significant role in the movement for Indian independence, there is no record of him introducing a bill for compulsory and free primary education.

Sir Harcourt Butler, option 3, was a British colonial administrator who served as the Lieutenant Governor of United Provinces (now Uttar Pradesh). There is no information to suggest that he introduced a bill for compulsory and free primary education.

Therefore, based on the provided options, Gopal Krishna Gokhale is the correct answer.