Final answer:
The member of Theodore Roosevelt's Cabinet whom he chose to run for President in 1908 was William Howard Taft.
Step-by-step explanation:
The member of his Cabinet that Theodore Roosevelt chose to run for President when he decided not to run for another term in 1908 was William Howard Taft. Taft was Roosevelt's Secretary of War and a close advisor. Roosevelt handpicked Taft as his successor, believing that Taft would continue the path of the 'Square Deal' that Roosevelt had initiated. Taft easily won the 1908 presidential election.