The middle initial in J.K. Rowling’s pen name represents Kathleen.
Rowling was born Joanne Rowling and did not have a middle name. Before publication of Harry Potter and the Philosopher’s Stone, her publisher asked her to use two initials because the market was thought to respond well to a gender-neutral author name. Rowling chose K from Kathleen, the first name of her paternal grandmother.
She therefore became J.K. Rowling for publication, with the J standing for Joanne and the K honoring Kathleen. The name is a pen name rather than a straightforward record of two given names received at birth.
This detail is often confused with Rowling’s later use of Robert Galbraith for her crime novels. Robert Galbraith is a separate pseudonym, while J.K. Rowling is the publishing name associated primarily with Harry Potter. Her legal and professional naming history also includes the surname Murray after her first marriage, but that does not change the origin of the K in J.K. Rowling.