It is not known for sure where the name for the game Chinese Whispers originated from. It is quite certain that the name originated from the British Empire, probably from England, and of course is an observation on the Chinese in one way or another.
Many people suggest that the name is a racial slur against the Chinese, implying that Chinese either gossip or lie a lot and are likely to twist and turn what you say into something else.
Other people suggest that the name was chosen for the game because when the English people hear Chinese being spoken as a language, it sounds like somebody mumbling and not speaking clearly. This is not such a surprising observation because the alphabets and the sounds of the languages are completely different and so it is very difficult for an English to hear distinctly different sounds in Chinese language, and the same is true the other way too.
Some people suggest that name comes because some people from Western countries often view China as being chaotic and disorganised and therefore use this name in the game to suggest chaos or disorder.
Finally, there are people who suggest that the name originates from observing that the Great Wall of China had communication posts spread all the way along it and that messages would be sent great distances along the wall and were therefore open to miscommunications and misunderstandings.
The vast majority of the rest of the world adopt the game under the name Broken Telephone Game, except for the French who also use a negative connotation towards arabs by calling their game The Arab Telephone Game, and the game was also once referred to as Russian Scandal game.
I tend to feel that the term must have come from an observation on either the way Chinese sounds to an English, or on the way it sounds to an English when a Chinese speaks English. This seems the mostly likely as it is a game of communication.