I was editing an article that referred to the Wörthersee, in Austria. As it stood, the copy referred to “Lake Wörthersee,” but a reader who speaks a little German might know that See is German for “lake,” and find that wording redundant.
A fellow editor who used to work for a news service in Japan told me that their house style was to replace the Japanese suffix for the geographical entity with the English word: Fujisan would become “Mount Fuji.”
The Chicago Manual of Style recommends leaving out the English term: “the Rio Grande” not “the Rio Grande River.” (Chicago Manual of Style, 16th ed., 8.54 “Foreign terms for geographic entities.”)
For the Wörthersee, I chose “Lake Wörther” instead of “the Wörthersee,” since a) it isn’t a very well-known place, so I didn’t think changing the name would cause confusion; and b) the German for “lake” isn’t that widely known. Based on the same considerations, I would leave Rio Grande, which is more widely known to North American readers, as is instead of changing it to “the Grande River.”
A similar problem comes up with some Scandinavian place names, where the name contains the definite article as the suffix “-et.” Referring to the city hall in Oslo, you have to choose between “have you seen Rådhuset?” and “have you see the Rådhus?” I’m not sure what the best solution is there.