“Come on up and I’ll show you my etchings” is an old veiled language joke that has been printed on many images. The speaker (usually a man) wants to lure a person (usually a woman) into a home or office not to display etchings, but to have a sexual encounter.
   
The comical pick-up line expression became popular in 1933, when it was used in print advertisements for the movie The Warrior’s Husband. “Come up and I’ll show you some of my etchings” was printed in an ad in the Akron (OH) Beacon Journal on May 18, 1933. “Come on up and I’ll show you my etchings” was printed in an ad in the Minneapolis (MN) Tribune on May 24, 1933.
 
“Want to come to see my etchings?” is dated and has declined in use. “Netflix and chill” is a modern expression from 2009.
 
       
Wiktionary: show someone one’s etchings
Verb
show someone one’s etchings (third-person singular simple present shows someone one’s etchings, present participle showing someone one’s etchings, simple past and past participle showed someone one’s etchings)
1. A clichéd innuendo used to offer to bring someone to a private location in order to have sex, to show a place where people can engage in sexual intercourse (alternatively replacing “someone” with a phrase starting with to)
Let me show you my etchings.
 
Wikipedia: Etchings
“Etchings” euphemism
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. Alger was an immensely popular author in the 19th century—especially with young people—and his books were widely quoted. In chapter XXII of the book, a woman writes to her boyfriend, “I have a new collection of etchings that I want to show you. Won’t you name an evening when you will call, as I want to be certain to be at home when you really do come.” The boyfriend then writes back “I shall no doubt find pleasure in examining the etchings which you hold out as an inducement to call.”
 
This was referenced in a 1929 (1939 is the correct year—ed.) James Thurber cartoon in which a man tells a woman in a building lobby: “You wait here and I’ll bring the etchings down”.
     
Wikipedia: The Warrior’s Husband
The Warrior’s Husband is a 1933 pre-Code American comedy film directed by Walter Lang and starring Elissa Landi, David Manners, and Ernest Truex. It tells the story of the Amazons, who ruled over men thanks to the sacred girdle of Diana, and Hercules who came to steal it. The film is based on a 1932 Broadway production of Julian Thompson’s 1924 play that starred Katharine Hepburn in the lead role.
 
Newspapers.com
18 May 1933, Akron (OH) Beacon Journal, pg. 23, col. 4 ad:
“Come up and I’ll show you some of my etchings”
(An advertisement for the movie The Warrior’s Husband.—ed.)
 
Newspapers.com
24 May 1933, Minneapolis (MN) Tribune, pg. 16, col. 8 ad:
“COME ON UP AND I’LL SHOW YOU MY ETCHINGS”
(An advertisement for the movie The Warrior’s Husband.—ed.)
       
Google Books
The Thin Man
By Dashiell Hammett
New York, NY: Alfred A. Knopf
1934
Pg. 5:
“And how about the redhead you wandered off with at the Quinns’ last night?”
 
“That’s silly,” I said. “She just wanted to show me some French etchings.”
 
Art.com
“Won’t you step in and look at my etchings?” - New Yorker Cartoon
By Rea Irvin
Published May 29, 1937 “Won’t you step in and look at my etchings?” Caveman to cavewoman as he invites her into his cave.
 
Google Images
Google Books
1 July 1939, The New Yorker:
“You wait here and I’ll bring the etchings down” (A cartoon by James Thurber, 1894-1961.—ed.)
 
Mic
Was This Old-School Meme the First Ever “Netflix and Chill”?
By Mathew Rodriguez
Oct 26 2015
(...)
One of the earliest known uses of “come up and see my etchings” that Mic found is from Horatio Alger Jr.‘s 1890 book The Erie Train Boy, which is partially available on Google Books. In a correspondence between a female character and a male suitor, the woman invites the man to view some etchings. “I have a new collection of etchings that I want to show you,” she wrote.
(...)
Though Alger’s inclusion of the phrase shows a woman inviting a man, in most subsequent iterations the genders are reversed. In this 1939 New Yorker cartoon parodying the phrase, a man invites a woman upstairs. In another 1937 New Yorker cartoon, a caveman invites a woman into his abode to view his cave wall etchings.
 
New York magazine
FEB. 16, 2006
Would You Like to See My Etchings?
By Em & Lo
P ity the poor suits—bedroom stereotypes have never credited them with much skill in the sack