The character of Hal King was named after Lucy's make-up man.
Luci Arnaz sings "Cumbanchera," one of her father's biggest hits.
In the speakeasy sketch, Lucy plays Dallas Noonan, doubtless a satire on Texas Guinan, one of the first female emcees. During Prohibition, she opened a speakeasy called the 300 Club in New York City. It became famous for its troupe of scantily-clad dancers and for Guinan's distinctive aplomb.
The title is from the Alan Jay Lerner and Frederick Loewe musical/movie My Fair Lady (1964) which was based on the George Bernard Shaw play Pygmalion (1938).
The characters played by Hal England and Larry Meredith were named after the "Here's Lucy" make-up artists, Hal King and Lee Greenway, respectively. Annie Whipple (as she freaks out (way out), goes psycho, tosses a couch pillow or 2, crouches down and just absolutely bunnykicks upon the Carters' front room couch): "A-A-H-H-H-H-H, HAL KING, A-A-H-H-H-H-H, A-A-A-H-H-H-H, OH, NO, HE WOULDN'T, HE COULDN'T, HAL KING, A-A-A-A-H-H-H-H-H!!!!" Uncle Harry: "You better fasten her seatbelt." (followed up by just several minutes later): "Forget the seatbelt, she needs a straitjacket!!" (also followed up by even that much later in season 5's 13th 1972-73 episode): "Well, girls, I think you still have a little work to do."