The “K” in J.K. Rowling’s name represents Kathleen.
Rowling was born Joanne Rowling, without a middle name. For publication, Bloomsbury asked her to use two initials because the publisher believed a gender-neutral author name might appeal to boys. Rowling chose “K” in honour of her paternal grandmother, Kathleen Ada Bulgen.
The name J.K. Rowling therefore combines her given name, Joanne, with a family name chosen for publishing purposes. She is often called Jo by friends and has also published crime novels as Robert Galbraith.
The initial is sometimes assumed to stand for Katherine because that is a common spelling of the name, but Rowling has identified Kathleen as the intended name. This is separate from the “K” used in other authors’ names and from characters in her fiction.