If I recall the term white elephant gifts refers to a gift given to Pres. Abraham Lincoln by the then King of Siam (don’t know if he was the same one that Anna met in the play) of, yes, a white elephant, which was a very precious gift not only because of the rarity of such an animal, but because only the king could own one. Abe thought, nice gesture, but where in the world are we going to keep it? I imagine it went to a zoo and lived in great dignity but I don’t know.
so know a white elephant gift is something one person might prize highly, but another person may not appreciate at all, and some parties that have a “white elephant” gift exchange expect participants to bring something they have been given, that has no value to them, but might be considered valuable or desireable by someone else. sometimes there is a draw from the hat, or sometimes a game is involved. You draw a number, you can either choose one of the unopened gifts, or you can take an opened gift from someone else, and they take an unopened gift. this game can be pretty hilarious, and a good ice breaker (do I need to explain that term?)