One famous example in such respect is the name of Pakistan, which however was coined purposedly when it became an independent nation. It comes from an acronym formed from the names of the five northern provinces of British India:
Punjab, Afghan Province, Kashmir, Sind, and Baluchistan (see here). The i vowel was inserted for euphonic purposes. However, the word paki also has its own meaning in Urdu ('pure').
Another such artificial creation is the name of Banzare Coast, coming from the name of the British-Australian-New Zealand Antarctic Research Expedition that discovered it (see here).
You would probably prefer natural developments to artificial creations. I am sure there are many examples, and I can mention those of two Russian cities. The first is Saint Petersburg, which is commonly known, and effectively called, Piter by the native speakers. Some people call it by the acronym SPb, even if more in writing than in speaking. Another example is the city of Yekaterinburg, which is commonly abbreviated into Ye-burg, also in the spoken language.
City names in the Spanish domains in South America were often derived from quite lengthy phrases having a religious content. Later they underwent spontaneous abbreviation. This is the case of Buenos Aires, a name derived from the former denomination Puerto de Nuestra Señora Santa María del Buen Ayre (at least, according to the popular sources).
P.S. I suppose I don't need to mention Frisco for San Francisco.