As I listened to coverage of the terrorist murders in India's financial capital (sadly, one rarely hears news from that part of the world unless it involves the loss of human life), I noticed that the correspondents referred to the city by its official name, Mumbai, while the vast majority of Indians called it Bombay, its former name. My son-in-law, who comes from a city nearby, calls it Bombay, as does Suketu Mehta, author of Maximum City: Bombay Lost and Found.
The Portuguese, who in 1534 appropriated the islands that would make up the city, called the area Bombaim, which may mean "good bay" or "little bay." When the English took over in 1661, after receiving the island as dowry for Catherine de Braganza, who married Charles II, they Anglicized it to Bombay. Hindi- and Urdu-speakers called it Bambai, while Marathi- and Gujarati-speakers called it Mumbai. In 1996 the government of Maharashtra renamed the city Mumbai, in an effort to remove colonial names. Mumbai is derived from the Hindu goddess Mumbadevi.
Other cities in India have had similar changes: Madras became Chennai in 1996 because Madras was believed to be a Portuguese name (a contraction of Madre de Deus), or more likely from the Madeiros family. Although Chennai predates Madras, quite a few of the residents still refer to the city by its old name.
Malaysia has also changed names--most notably Jesselton, in Malaysian Borneo, which was renamed Kota Kinabalu--literally, the city of [nearby] Mount Kinabalu--in 1968. I suspect the new name has been more accepted there, but Malaysian singer Pete Teo's hit, "Jesselton Tonight," uses the old name. But its line "Would you fall in love in Jesselton ere days of 'burn baby burn' hearkens back to earlier days. "Burn baby burn" refers not to H. Rap Brown's slogan of the 1960s but to the destruction of forests for agriculture.
Other countries have been much more successful in getting new names to stick. I don't expect Zimbabwe to revert to Rhodesia after Robert Mugabe is out of power. Nor will Kinshasa go back to Leopoldville. If a name represents a despotic government, it usually will be replaced. Leningrad is now St. Petersburg, thus replacing the name of one despot with that of an earlier one (though Tsar Peter the Great had the modesty to name it for the saint who shared his name). Tsaritsyn became Stalingrad in 1925, but Nikita Khrushchev renamed it Volgograd in 1961. I suspect that someday Myanmar will again be Burma. In fact, virtually all the opponents of the despotic Myanmar government call the nation Burma.
But the most successful sub-Saharan African country, the Republic of South Africa, has kept Johannesburg, Cape Town, Durban, etc. , even though they're reminders of the colonial past. Maybe they've decided that changing the name doesn't change the reality.
But if a name change is to be truly successful, it needs to be supported by the people. Had India changed Bombay to Mumbai in 1947, in the days of independence, the people might have embraced it. But to change the name a half century after Indian independence and without any popular referendum made no sense. But then, the expense of changing it back may make even less sense. There's the old joke about the expense of changing Boulder Dam to Hoover Dam--that it would have cost less if Hoover had changed his name to Herbert Boulder. So I suspect that Mumbai will continue to be the official name of India's largest city, while its residents will keep calling it Bombay.