It's a given that place names are loaded with political significance, and that conquering armies and factions will rename cities and even entire countries to suit their ideological goals or personal whims. Thus did the Soviets rename St. Petersburg to Leningrad to honor their own founder, and no sooner did the Communist regime collapse than the newly liberated Russians reinstated the city's original name, honoring a tyrant whose rule the passing centuries had made seem less cruel than that of Lenin and Stalin. The Irish town of Derry, renamed Londonderry after its conquerors' capital, carries even greater nominal weight; locals will judge a visitor's political stance by which name he uses to refer to the place. Exiles and refugees can denounce the current rulers of their former homes simply by keeping old names. A professor I studied with as an undergraduate fled Iran many years ago at the outbreak of theocracy; now a well-known researcher sending American satellites into orbit, he still defiantly proclaims himself a Persian.
But there's more to it than simply the name of a place. In some cases, how you pronounce a place's name can be as important as the name itself. Growing up in the vicinity of Baltimore, I knew plenty of people who spoke in that city's uniquely consonant-depleted drawl. My mother, the first in the family to attend college, regarded such pronunciation as uncouth, and cultivated a style of speech so flat and dictionary-perfect that her neighbors asssumed her to be a recent arrival from the Midwest. Lately, though, Baltimorese has regained popularity among the city's better-educated residents, partly (I suspect) as a tongue-in-cheek political statement against those Washingtonians bent on turning all of Maryland (and much of Virginia besides) into a suburban amusement park.
Being situated on the Atlantic coast, I inevitably passed through the Appalachian Mountains any time I traveled westward. Everyone I knew, Baltimorean or otherwise, pronounced the range's name "app-a-lay-shun" with emphasis on the long central "a." I said it that way, as did a roommate from Richmond who talked of rafting in the region's rivers, and a neighbor from Pittsburgh who had gone hunting there. It never occurred to me that such a pronunciation might not be universal.
A few days ago, though, here in the Rocky Mountains, I was talking with a person originally from Johnson City, Tennessee, comparing the Rockies to the Appalachians back east. He repeatedly corrected my pronunciation, insisting that the correct way to say the shorter range's name was "app-a-latch-en" with the sharp "a" stressed. I replied that I'd never heard it said that way by anyone, at which point he made the analogy to Derry/Londonderry - you can say it either way, and the locals will understand you, but if you say it their way they'll like you better, and won't give you wrong directions or shoot at you.
My Appalachian colleague was not offended by my ignorance of proper pronunciation (though he did have a few odd things to say about Baltimore accents). I was a bit shocked that such a minor variation in speech could have such great implications on interpersonal relations, though. The current Webster's dictionary gives my pronunciation of "Appalachian" as the preferred one and the local variant as an alternate; apparently people who actually live in those mountains don't have much of an influence with lexicographers easily swayed by popular consensus. I note, however, that New Yorkers' speaking of their city as "New Yawk" doesn't even merit mention in the dictionary, while a "local" pronunciation of Baltimore does.
Update: VT_Hawkeye adds that either pronunciation is fine for the adjectival form "Appalachian," but the region Appalachia is properly pronounced "App-a-latch-uh."