I'm asking in the context of medicine, where you're "massed" (or is it "weighted"?) on a scale.
This answer beneath insinuates that 'mass' may be the preferred term, but it doesn't outwardly clarify if 'body weight' is wrong and ought be replaced by 'body mass'.
So why don't we call it "mass"? Well, according to http://www.etymonline.com/index.php?term=weigh, "weight" is a very old word,
The original sense was of motion, which led to that of lifting, then to that of "measure the weight of." The older sense of "lift, carry" survives in the nautical phrase weigh anchor.
Before Newton, the concept of inertia didn't exist; so the distinction between mass and weight made no sense when the word was first introduced. And we stuck with it...