C.S. Lewis’s childhood nickname was Jack. He adopted the name after a childhood dog named Jacksie was killed; Lewis then used Jacksie, shortened to Jack, as his own name.
Lewis was born Clive Staples Lewis, but people close to him generally called him Jack for the rest of his life. The nickname was used by his brother Warren, his friends, and later colleagues at Oxford and Cambridge. It became the familiar name associated with his personal correspondence and biographies.
The name C.S. Lewis came from his initials rather than from the nickname. He published fiction, literary criticism, apologetics, and children’s fantasy under those initials. Readers sometimes assume “Jack” is a pen name, but it was primarily his personal name among people who knew him.