The United States Board on Geographic Names is the government body that has the power to change place names of locations within the United States. It also has the ability to change government policies of foreign places names. For instance, when working with the State Department the board recognized the "Republic of Macedonia" instead of "The Former Yugoslav Republic of Macedonia." It also changed American recognition of the spelling of Ukraine's capital from the Russified Kiev to the Ukrainian Kyiv.
The US Board though is not the only geographic naming body around, though. According to the United Nation's list, there are many geographical bodies out there (PDF). There is also a contact list in case one wishes to visit said country's board website or contact them via mail (PDF).
There are a few surprises. Centralized countries who take deep pride in their geographic terrain like Mexico, Ethiopia, and Egypt do not have a geographic name board while Afghanistan does.
Just because a country has a geographic board does not mean it has the final say over place names. Russia's Commission of Geographical Names had no input when Prime Minister Putin named a mountain in the ethnic republic North Ossetia-Alania the Peak of Russian Counterintelligence Agents. Meanwhile, I can only image how many times North and South Korea's geographic boards disagree. Finally, while the Vatican does not a geographic board the UN notes their hope that the Holy See will have one soon. I do not know what they could rename but I want on it! (Hat Tip: Hunter)