|The Bing home page varies its look with each refresh - screenshot via Wikimedia|
Microsoft was developing a search engine to compete with Google and others and wanted a name that was short, memorable, and easy to spell or rather not misspell. Their previous attempts at a search engine had been called MSN Search, Windows Live Search and later Live Search.
Before settling on "Bing," they had supposedly considered "Bang.” That was rejected because if it was used as a verb (as in "Googling" something), the resulting "I Banged it" sounded inappropriately obscene.
So, they went with “Bing” which met their requirements. It also suggested the term sometimes used when someone finds something they were looking for - bingo!
I also see mentioned that when a lightbulb goes off over a cartoon character's head(a "lightbulb moment") to mean they just got a good idea, you often hear a “bing” sound effect.
In China, the Bing website is called bì yìng, which translates as “very certain to answer.” That sounds good too. Bing's detractors have erroneously suggested that it is an acronym for Bing Is Not Google.