C.S. Lewis’s surname was Lewis; “Jack” was a nickname he adopted for himself as a child.
Lewis’s full birth name was Clive Staples Lewis. As a young boy, he disliked the name Clive after the death of his dog, Jacksie. He began referring to himself as Jacksie, later shortened to Jack, and the nickname remained with him through adulthood.
This question concerns a nickname, not a change of surname. Staples was Lewis’s middle name, taken from his mother’s family, while Clive was his first given name. “C.S.” refers to Clive Staples, the initials he used as an author.
Friends and colleagues commonly called him Jack. Readers, however, usually encounter him under C.S. Lewis, the professional name attached to his scholarship, fiction, and Christian writing.