Nick Bilton of the New York Times was the first to reveal the nickname:
A source at Apple told me while they were making the gold iPhone it was referred to internally as "The Kardashian Phone."
TechCrunch followed up with its sources to confirm:
Apparently this was not a serious ‘code name’ or anything, just an internal joke. Kim Kardashian, of course, has a well-known connection to gold iPhones. The reality star actually had an iPhone 5 that was anodized gold long before Apple’s ‘champagne’ device hit the market, as you can see in the image below.
It was BGR that arranged for two gold anodized iPhone 5 devices to be sent to Jonathan Cheban and Kim Kardashian which boosted the popularity of the color. However, BGR has checked and Kim is using a black iPhone 5 now, and Jonathan is using a silver 5s and tells them, “Gold is so tacky now! Been there done that!”