Give somebody a hard time

Meaning

To give somebody a hard time has two faces, and you need the context to know which one you are looking at. The friendly face is teasing: your friends give you a hard time about your new haircut, and everybody laughs. The serious face is making someone's life difficult: criticizing them, arguing with them, or refusing to help. A customer can give a waiter a hard time. A boss can give a new hire a hard time.

The picture is simple: you are handing someone a difficult time, like a package they did not ask for. It is informal and very common in American English. A tone of voice usually tells you which meaning is in play. "I'm only giving you a hard time" is how you signal that you were joking and you did not mean to hurt anyone.