Btw, the Korean name "baduk"/"badook" is also used quite often. Sadly Chinese version is the least well known, which I think is because China and USA are the least connected of the three big Weiqi countries.
The complete word is actually "Igo", using characters not too different (maybe traditional?) from the characters for Weiqi. Together Igo/Weiqi means "surround + chess-like". The last character Qi/Go refers to the corresponding more famous chess-like game, so in China it's Xiangqi, but in Japan it's Igo/Weiqi.
I have always called it Weiqi as I was taught in mandarin. From what I understand, Weiqi originated in China, and for most of history was the dominant player in this game, until Japan took over around World War 2 (the rise of Japanese military power , technology, and standards of living probably all contributed to this).
The Japanese took up the game, called it Go and helped spread the game to the west and the name has stuck ever since.
Japan has since been eclipsed by both Chinese and Korean players though.
International Go Tournaments are now mainly dominated by Chinese and Korean national team players it seems to me.