Proper nouns with compulsory articles included loosely follow a few patterns, but are a matter of case-by-case determination, and often case-by-case debate. In the end, the rule for each name simply needs to be memorized, and no matter how thoroughly you do so, expect to be disagreed with at least some of the time.
One type of proper noun that might include "the" is a place name that derives from a description. The United States of America, The British Isles, The Netherlands, and The Upper West Side are examples of this. (Which lands? The nether ones!) When a description graduates to the status of a proper noun, the article does not automatically fall away (though it can). Some names that derive from descriptions are not candidates for included articles at all. One such is Martha's Vineyard, simply because the article doesn't fit with the descriptive reading of the name in the first place. To play off the above example of "He Who Must Not Be Named" we also have the moniker of "the dark lord", which is no way a proper noun, but what if it became one? Not necessarily likely since there were other historical dark lords in the Potter universe, but conceivable, as important figures in history will eventually need a name that reporters aren't too afraid to put in the paper. (Few names could be more confusing than You-Know-Who. "You-Know-Who was just here!" "No I don't.") So supposing it did become a legitimate name (proper pseudonym?) for Voldemort, would the name be "The Dark Lord", or "Dark Lord"? It could really go either way according to preferences at the time. If people were already accustomed to hearing "the dark lord", anything else might be jarring. And like every other similar case, once the convention was set, everyone would just have to learn it, or else send protesting editorials to the Daily Prophet.
Another type of proper noun to potentially start with "the" is anything named after a river. While "the" is not generally considered to be part of a river name, it is almost always included when using a river name in a sentence. For example, the map reads "Nile River" but we talk about "the Nile River". (Note that the article need not be capitalized as it's not part of the name.) The Bronx, mentioned above, is a good example of that. I suspect that The Congo (often referred to as simply "Congo", and either one is still an abbreviation), is an example as well. (According to Wikipedia, The Congo was named for the Congo River, which was in turn named for the pre-colonial Kingdom of Kongo, which itself was named for its first ruler.) Without any actual statistics on this point, I would hazard a guess that most things named for rivers do not inherit the article.
And, I suppose any time someone is deliberately naming a thing, they can choose to include an article if they like, though I honestly can't think of any good examples of people doing so. In the case of Facebook, had Zucherberg chosen to name the site "The Facebook", this would have emphasized the literal description in the name, the book of faces. If he had done so, I think it would have weakened the branding of the site, and associated the site more closely with the idea of "the book of faces", which would likely have caused many users to be less accepting of the expanding function of the site. (Though that's entirely off-topic.)
So, once a proper noun includes the definite article, its ongoing use can still be challenged, and this isn't uncommon in country names. A good example of this is Ukraine, which was generally referred to as The Ukraine before the fall of the Soviet Union. By now, most English speakers have switched to using "Ukraine", though you can still find some hold-outs. For a period of time during the transition, you could easily have found someone to tell you you got it wrong regardless of which version you used.