It was a mistake.
Out of universe, just a production goof. In universe, Gary Mitchell made a human error.
From Memory Alpha:
The gravestone Mitchell creates for Kirk says "James R. Kirk". According to D.C. Fontana in the introduction for Star Trek: The Classic Episodes 1, when the mistake was discovered, Gene Roddenberry decided that if pressed for an answer on the discrepancy, the response was to be "Gary Mitchell had godlike powers, but at base he was Human. He made a mistake."
Additionally, from IMDb's trivia page:
According to the Starfleet Access commentary on the Blu-Ray, the remastering crew debated over whether or not to change the middle initial on the "James R. Kirk" tombstone to the proper T. While some members of the crew were for it and some against it, they ultimately decided not to, due to the ridiculous amount of rotoscope work it would have required.