White elephant is one of many different names for the gift exchange game where people get to swap or steal what others have opened. I've also heard it called a Dirty Santa or Yankee Swap (derived from the hard-core frugality associated with old New Englanders).
I've always thought of a white elephant as something useless or superfluous, the way MP's dad uses it to refer to the stadium. I've also heard the term "white elephant sale" (the implication being that people are selling their useless/superfluous-to-them stuff), but it's been at least 40 years. These days everyone I know calls them rummage, garage, yard or tag sales. (IME, churches/institutions have rummage or tag sales; families have yard or garage sales.)
Speaking of which, what other names do (general) you call Secret Santa exchanges, where people draw names and give gifts anonymously to the person they picked? Over the years I've heard them called Christmas angel, Kristkindl (German speakers, please forgive my spelling), Kriss Kringle and -- this one really made me go "huh?" -- Pollyanna.