The correct answer is option 3, James Mill.

Option 1, Charles Grant, is not the correct answer. Charles Grant was a British politician and philanthropist, but he is not known for writing the book `History of British India`.

Option 2, John Stuart Mill, is also not the correct answer. John Stuart Mill was a philosopher and economist, and while he did write about British India in some of his works, he is not the author of the specific book mentioned in the question.

Option 4, William Jones, is not the correct answer either. William Jones was a British judge and philologist who is known for his contributions to the study of ancient languages, but he did not write the book `History of British India`.

Therefore, the correct answer is option 3, James Mill. James Mill was a Scottish historian, economist, and political philosopher who wrote the book `History of British India`. In this book, Mill provided a comprehensive account of the history, administration, and politics of British India.