In modern Dutch, the name of the city is either 's Gravenhage ("The Count's Hedge/Terrain/Court") or Den Haag ("[In] The Hedge"). Haag is the same word as -hage. The name Den Haag is the neutral or less formal name, whereas 's Gravenhage is rather formal or official.
The origin of the name is that the court of the Counts of Holland and their successors has been in The Hague since the Middle Ages and still is (now the Royal Court and Parliament); one would say "hij is in den Hage", meaning "he is in the Hedge/Court", where den is the dative/accusative case of the article de, "the". So it would have made more sense for the name to be simply Haag. But the inflected article indicates by its case that it is about a location, and it was somehow coupled with the proper name and has become part of it.
In many cases, such an article is lost in translation, because articles are usually not used the same way in a different language. But, in this case, other languages kept and translated the Dutch article, so in French it is La Haye, in Spanish La Haya, in English The Hague.
The city of Istanbul probably (this is not entirely certain) comes from Greek Eis tên polin, meaning "(In)to the city". It was somehow interpreted as a proper name and written attached, and other languages just started using the Turkish name at some point.
The French city of Lille comes from l'Ille or l'Isle, meaning "the island". But the region it is in used to speak more Dutch than now, and the official name in Dutch is now Rijsel, which comes from Ter IJsel, "In(to) the Isle", which was incorrectly analysed as coming from *Te Rijsel. French and Dutch still have many other modern city names having definite articles and even prepositions, like Le Havre ("The Haven"), Den Ham, Ter Apel ("in/at/to the Apel"), etc.