[glimshe]

Do electors of any politician ever understand the full extent of their candidate's powers? They vote on people under the assumption that the person is the best choice (not necessarily a "good" choice) for the job given the system already in place.

Answering your question: yes, the executive branch can change geographical names in the US. Obama did that too with Mt Mckinley. This power extends solely to the names as accepted by the United States. If Mexico wants to change the gulf's name to "Gulf of the Aztecs" or "Gulf of qetwyetwdhxysysheussg", it can do that too.

SHOULD they do it? Now that's a different question.