As I'm sure you know, weight is mass times gravitational acceleration so it doesn't make sense to talk about the weight of the Earth unless you were putting the whole Earth in some gravitational field.
I would guess in that phrase the word weight should be mass.
I was studying mechanics kleppner and kolenkow and in 3rd chapter it said that Cavendish became famous for finding G and people started calling him man who "weighed the earth" @JohnRennie
Ya we are measure our weight wrt to earth so it is meaning less
Ah, yes, I have heard that before. I think it's just because there is no verb meaning to measure the mass like the verb weigh means to measure the weight.