Mumbai vs Bombay

The name Mumbai is an eponym, derived from Mumba or Maha-Amba—the name of the Koli goddess Mumbadevi—and Aai, “mother” in the language of Marathi.

A widespread popular etymology of Bombay holds that it was derived from a Portuguese name meaning “good bay”. This is based on the facts that bomis Portuguese for “good” and baía (or the archaic spelling bahia) means “bay”. However, this literal translation would have been incorrect in grammatical gender, as bom is masculine, while baia is feminine; a correct Portuguese rendering of “good bay” would be boa ba(h)ia. Having said this, baim is an archaic, masculine word for “little bay”.

By the late 20th century, the city was known as Mumbai or Mambai to Marathi and Gujarati speakers and as Bambai in Hindi, Persian and Urdu. The English name was officially changed to Mumbai in November 1995.  A theory was proposed suggesting that “Bombay” was a corrupted English version of “Mumbai” and an unwanted legacy of British colonial rule.

- Wikipedia