Hi Tura, for the sake of someone highlights the fact that you are doing or saying something to help or bring advantage to someone. (i.e. Please do it, for Mary's sake or I hope he's there for John's sake). You could use "for" as well, but the sentence wouldn't get the same emphasis.
For the sake of something underlines the purpose, the reason why you are doing or saying something. A good synonym could be "because of".
I hope I have been helpful.