Ask any entity which has been through the wringer with a whopper of a lawsuit. And you will find out that it had to struggle against being preoccupied with the litigation. That is, not getting distracted from the business of their business.
So, the primary motivation for Donald Trump to file a lawsuit over the eligibility of Ted Cruz to run for the U.S. presidency could be to throw the candidate and his campaign team.
In addition, litigation can be a platform for publicity. In America, suing is sexy. Tabloids such as the New York Post cover that kind of action all the time.
Sure, a lawsuit is expensive. But, Trump has the resources to keep that up forever.