Should I mention my goal of expanding my family business in my MS statement of purpose? I am applying for MS in Industrial Engineering. Do you think it is safe to mention that I have a family business which I would like to expand with the aid of IE? Or will it be useless for their purposes since they may perceive me as a rich kid with back up who does not genuinely have the want and need to study?
In my opinion, having a clear reason for going to graduate school can only help your application.  Many people apply to graduate programs simply because it is the next thing after undergraduate.  For you, there is a clear motivation and a source of focus.

With regards to this particular reason, it is not clear to me whether it would be good for a Ph.D., because of the research component.  A master's degree, however, is designed to give you knowledge and skills that can be applied outside of academia, however, and so your motivation of applying this to a particular business is a good match (the fact that it is a family business I think should not particularly matter).