Why the name change from Peking to Beijing?

Generally, it was to keep the name in line with the significance of it being a capital.

https://en.wikipedia.org/wiki/Beijing#Names

The Chinese capital did not change its name but Chinese words became spelled in English differently. In Chinese, the name stayed exactly the same and most Chinese people are not even aware that some Westerners think that there has been a name change. The old spelling has been Peking, this is how the city appeared in most earlier discourse. After the establishment of the People's Republic of China in 1949, the government adopted the pinyin transliteration method and used this to write all of the proper names (including place names, people's names, etc) using the Latin alphabet. Theoretically, this was when Peking became known in the West as Beijing. In reality, however, the West has been using the old spelling long after it has been replaced in China. It is only sometime in the 1980s that China started to enforce its official name on all flights, sea routes and official documents. This is why the name Peking is still echoing in our minds and people continue to use it even today. Needless to say, it is easier to pronounce than Beijing, which is an important factor too.

Both the old and new spelling are approximations of the Chinese sound, pronounced something like "pay-cheeng."

http://english.visitbeijing.com.cn/a1/a-X9Y2KW83538BAD8903653F

I might add: for many years, western English speakers used a transcription system called Wade-Giles to render Chinese words and names. Once we began to adopt the pinyan system mentioned on zbeck's site, many such words appeared to change:



https://en.wikipedia.org/wiki/Wade-Giles

Peking is the name of the city according to Chinese Postal Map Romanization, and the traditional customary name for Beijing in English (passports issued by the British Embassy are still printed as being issued by the "British Embassy, Peking"). The term Peking originated with French missionaries four hundred years ago and corresponds to an older pronunciation predating a subsequent sound change in Mandarin from [kʲ] to [tɕ][10] ([tɕ] is represented in pinyin as j, as in Beijing). It is still used in many languages.

https://en.wikipedia.org/wiki/Beijing

As you say, not all countries changed from Peking to Beijing.
For example, it's still Peking in German, Hungarian, Croatian and probably some more languages.

http://de.wikipedia.org/wiki/Peking