What's the origin of the following saying? "This is the day they give
babies away with a pound of butter."

This line comes from an old song. Apparently the original version had
"half a pound of tea" instead of "a pound of butter." "Half a pound of
cheese" is also sometimes mentioned. According to some sources, the
song may date to the early part of the Twentieth Century. It was
widespread by the 1920s.

"This is the day we give babies away
 With a half a pound of tea
 You just open the lid, and out pops the kid
 With a twelve month guarantee.

'The Day They Gave Babies Away,' a story by Dale Eunson that appeared
in the Christmas 1946 issue of Cosmopolitan, their most successful
Christmas story ever. It was published as a book the following year.
The reference also mentioned a soldiers' ditty circulating in the
1940s that went 'Today is the day they give babies away / with a half
a pound of tea. / If you know any ladies who want any babies / Just
send them around to me.'

I can add references in Vance Randolph's Roll Me in Your Arms:
"Unprintable" Ozark Folksongs and Folklore (Univ. of Arkansas Press,
1992) and Ed Cray's 2nd edition of The Erotic Muse (Univ. of Illinois
Press, 1992), which mention its inclusion in a Josiah Combs 1925 book
of Kentucky/West Virginia folk songs and G. Legman's recollection of
the song in Scranton ca. 1925. A Google search revealed a number of
interesting recollections and usages (including another short story)
of the song and title, one or two of which may be earlier than 1925,
possibly from the turn of the (previous) century."

from Sing Out! The Folk Song Magazine: The Songfinder
http://www.findarticles.com/p/articles/mi_m1197/is_1_48/ai_113887014

Here you'll find a long thread discussing the song:

The Mudcat Cafe: Today's the day we give babies away
http://www.mudcat.org/thread.cfm?threadid=50412

My Google search strategy:

Google Web Search: "day * give babies away"
://www.google.com/search?hl=en&q=%22day+*+give+babies+away%22

I hope this is helpful. If anything is unclear or incomplete, please
request clarification; I'll be glad to offer further assistance before
you rate my answer.

Best regards,
pinkfreud

It  is a clearly written, well-researched answer with links for more information!