Gulf of America

If the renaming was motivated solely by political reasons, to relate a region with US rather than Mexico, then the reason given about representing South, Central and North America is inauthentic. In this case, it's rationalization . 
Also, the other person's response completely ignores  your argument about the historical name of the region (which is accepted worldwide for centuries), and offers only his reason for the renaming. This reminds kind of avoiding the issue , but on the other hand, one could say that for the other person, his reason for renaming the Gulf of Mexico is just valued at the highest point, so all other reasons for not renaming it, don't matter, therefore there is no need to be refuted. But, in order for an argument to be persuasive, it should explain WHY one rationale is superior to the others, so the other arguments should be addressed and compared with what one proposes instead. And since one proposes something to be changed, he should at least explain why that change is necessary or how that change is beneficial by solving current problems that might exist. 

I think it's a bit hard to pin a logical fallacy on this one, as the two people are debating their opinions on what the Gulf of Mexico should be called, rather than any sort of logical truth. However, I do see a possible red herring as the second person basically ignores the point about the historical context of the name, and gives a totally unrelated justification for why it should be called Gulf of America. That said, the justification isn't that good. The Gulf of Mexico only touches the shoreline of the US, Mexico, and Cuba, which are all part of North America, and the part that Trump renamed to Gulf of America is only the section that's along the shore of the United States of America, so the idea that it represents those other regions makes no sense. In fact, the justification that Trump gave for doing it was all about the United States. The idea that geographic features should be renamed to represent wider areas is not one I can recall ever having seen applied before. The fact that this justification is seemingly pulled out of nowhere might make it an ad hoc rescue .

No fallacy. And it makes a valid argument.