Why are they called Hudson Bay and Bay of Biscay? Why Mediterranean Sea and Sea of Japan?
I was looking up a French town on Google Maps, when it struck me. There are bays shown as "Bay of ..." on the map, as well others listed as "... Bay". Their naming seems to be consistent with the naming in other maps I checked.
Likewise we have most seas listed as "... Sea" (e.g. Caribbean Sea), but some oddballs like the Sea of Okhotsk. This also can be noticed in the naming of gulfs.
I simply can't figure out the rule. Here is a link to the list of seas on Wikipedia.
(On a random note, this is probably coincidental, but the names of bays I can think of off the top of my head that end in "Bay" are all in America.)
Solution 1:
There is not so much a firm rule with proper place-names as there is traditional nomenclature. While one can conjure a rule, even the most accurate of such would still likely have some exception, in either geography or fiction.
With that said, the example you note all infer a common rule for their construction: Bodies of water that are named for an adjacent locale are 'of' that locale, while those named for themselves are not.
The Hudson Bay and seas Mediterranean and Caspian are named for themselves. There are no adjacent locations named Hudson, Mediterranean, or Capsian.
The Bay of Biscany is adjacent to Biscany, while the Seas of Japan and Okhotsk are adjacent to Japan and Okhotsk.
Again, this is only an inferred rule, and you will likely find exceptions that follow some other naming convention if you look hard enough.
Solution 2:
Captain Cook named a bay in Australia 'Botany Bay' for its abundant plant life at the time.
Euphony within a language and translation from another language may also influence the form of a name. In Cuba there is a bay whose Spanish name is Bahía de Cochinos, and in English this is rendered "Bay of Pigs." It would be possible to translate this as "Pigs Bay" (interestingly, this does not refer to farm animals, but to a type of local fish), but a preference for the longer form may emerge among English speakers because of the strong trochee sound of "Bay of Pigs."