easy
The Pope has it but he does not use it. Your father has it but your mother uses it. Nuns do not need it. Arnold Schwarzenegger has a big one, Michael J. Fox's is quite small. What is it?
Common Wrong Answers
“Title”
While the Pope has a title, it is not something that is uniquely personal like a last name. Titles are also often used regularly, contradicting the riddle's clues.
“First name”
The riddle specifically refers to the last name, not the first name. The clues suggest a distinction between what the Pope has and how it is used or not used by others.
“Job”
A job is not something that can be owned like a last name. The riddle indicates a personal attribute rather than an occupation.