False, for reasons others have already pointed out.
Why do people believe this myth? Microsoft has plenty of Chinese employees and Qi Lu was the executive VP of Bing for quite a few years. If there was an issue with the name, it would have been addressed by their marketing department long ago.
It also means other things like "ice", but "sick" is the first thing that comes to mind, and in China this is especially a big deal due to superstitions. Naming things is very important here.
amusing anecdote: I remember a discussion about someone who wanted to name his shop 森林木 (three trees symbol, two trees symbol, one tree symbol) and everyone told him it was a bad idea and it would fail because the trees where getting fewer and fewer and that this name was an omen of bankruptcy, and if anything he should put it in the opposite order. (He ended up choosing a different name entirely)
Bing is a really bad name, even tho this had probably nothing to do with it's relative failure
No, "bing" by itself does not mean sick. The sound "bing" maps to at least 59 possible characters across the 4 tones[0] of Mandarin. Only one particular character that maps to "bing4" means sick and it's one out of at least 21 possibilities[1], though I'll grant that it's the one that people are most likely to think of first.
Moreover, as it's normally pronounced by westerners, Bing would most likely map to "bing1" anyway.
Why do people believe this myth? Microsoft has plenty of Chinese employees and Qi Lu was the executive VP of Bing for quite a few years. If there was an issue with the name, it would have been addressed by their marketing department long ago.