One approving comment came from the official account of Chicago The Musical, which commented: 'We're so flattered! Welcome to the world, Chicago West!'
Others were not so impressed, with Emma Deeley joking: Coming to McDonalds soon... the Chicago West burger!'
With its distinctive ring, Chicago West brought to mind "an apartment building" for Kelsey Renfro, "a community college" for Travon Free and "a hit new ABC medical drama" for Meredith.
Still more debated alternative names for the child.
"That was the perfect opportunity for Wild West and they blew it!" wrote Kellie, while Twitter user Hope reflected that the baby should have been named Philadelphia, in reference to The Fresh Prince Of Bel Air's intro.
Another fan, Shawn Carlow, linked the name to the 1959 Alfred Hitchcock movie North By Northwest.
He wrote on Twitter: 'Still the Kimye kids are following the North by Northwest naming system. Chicago is an important location in the film. Saint is the lead actress. North West, well, obviously.
"I hope they name their 4th kid Mount Rushmore or Crop Duster or something."
Kim announced the surprise choice of name on her blog on Friday morning, in the form of a birth announcement.
The card also carried the baby's January 15 birth date, alongside her birth weight of 7lbs 6oz.
The name is sure to hold deep meaning for the reality star's rapper husband Kanye West, who was raised in Chicago.
And the newborn already has a nickname - thanks to auntie Khloe.
Kim's little sister revealed that her niece was known as Chi, pronounced shy. "I loooooover her name," wrote the proud auntie. "Hey Chi (shy)".
Meanwhile a Seattle food truck named Chicago West has announced it will be releasing a new food item named in the baby's honour.
Owners Dave and Mellissa Chester joked to celebrity website The Blast: "We at ChicagoWest are so honored that Kim and Kanye would consider naming their child after our food truck.
"We didn't see them at the truck, but feel this must be a nod to our amazing Chicago Beef Sandwich.'
Meanwhile, many fans have claimed Rob Kardashian tweeted about the newest West family member's name on Wednesday.
In a single tweet, the younger brother of Kim typed up two bear face emojis.
Twitter users tried to figure out the meaning behind the tweet and some related the tweet to sport's teams in Chicago.
Many fans referred back to Rob's tweet after Kim and Kanye's name announcement - and some questioned if it was an early clue to the public.