This phrase originated with Emma Lazarus’ 1883 poem that has come to represent the voice of the Statue of Liberty, “The New Colossus“:
Not like the brazen giant of Greek fame,
With conquering limbs astride from land to land;
Here at our sea-washed, sunset gates shall stand
A mighty woman with a torch, whose flame
Is the imprisoned lightning, and her name
Mother of Exiles. From her beacon-hand
Glows world-wide welcome; her mild eyes command
The air-bridged harbor that twin cities frame.
“Keep, ancient lands, your storied pomp!” cries she
With silent lips. “Give me your tired, your poor,
Your huddled masses yearning to breathe free,
The wretched refuse of your teeming shore.
Send these, the homeless, tempest-tost to me,
I lift my lamp beside the golden door!”
In the modern expression, give has become bring.
The huddled masses are still most common in this phrase, followed by criminal masses1, hungry, and then there a few variations that don’t quite fit the “immigrant” frame set up by this quote: old sewing machines [pdf], dirty bombs. So it would seem that variation in this snowclone is not as wide as in some of the others I’ve covered–the X really is mostly limited to the set of things (people) that can at the same time be described as “poor” and “tired”. This list of descriptors may not be as tangentially related to each other as X, Y and Z sometimes are in “X and Y and and Z, oh my!”
1 Two snowclones for one on that blog post (the other being “we don’t need no stinking X!”), though I don’t recommend you click it.
The Snowclones Database 
“Colossus”, please. One L, two S’s.
Holy crap, do I not proofread at all? For shame. Fixed!
I used this while working as the stick man on a game of casino craps. Before a point is established, the stick person advertises the bets in the center of the table, such as the horn or the yo. I would sometimes say, “give me your horns, your yo’s, your center bets who yearn to breathe free.” I thought I was being original. Guess not. 🙂
That’s the first instance of that variation I’ve heard, so I’d say it’s pretty original. 🙂