“For your own sake, I really hope they don't cancel.”

You are a manager in a sales department. One of the salespeople that works under you made a mistake with a major client, and now the client is now threatening to cancel their account with your company. You are speaking with the salesperson and want to suggest that you will be able to forgive the mistake if she can convince the client to stay, but that you may have to fire her if they cancel. You say:

For your own sake, I really hope they don't cancel.

for (someone's) sake

This phrase means "for (someone's) benefit or good". You use it to express why you're doing something. You usually use this phrase when you're describing doing something large and important to make a person happy or keep them safe:

Elizabeth decided for her mother's sake not to get a tattoo.

"For your own sake" is used to give people advice or, as in the example above, to threaten them. You use it in the sense "I hope something happens for your own sake" or "Do something for your own sake."

I hope this deal doesn't fall through, for your own sake.

Please consider getting life insurance, for your own sake as well as your family's.

I really hope (relative clause)

You "hope" something is going to happen when you want something to happen, but you're not sure yet if it will happen or not. This is different than "wish", which is for things that are impossible or probably not going to happen.