The phrase “Want to come up and see my etchings?” is a romantic euphemism by which a person entices someone to come back to their place with an offer to look at something artistic, but with ulterior motives. The phrase is a corruption of some phrases in a novel by Horatio Alger Jr. called The Erie Train Boy, which was first published in 1891.
I always liked this one, for the implication that something as cool as etching was once a common passtime. I guess the modern equivalent may be “check out my 3d prints”, but it feels nerdier.