Botn/botten is cognate with the English word bottom, and it might be part of a general north European distinction of lowlands, as opposed to highlands, such as the Netherlandic region, Samogitia (Lithuanian), and Sambia (Russia).
A second possibility is that botten follows an alternative Scandinavian connotation of 'furthermost'. Thus, the Gulf of Bothnia would be the farthest extent of the Ocean.
Julius Pokorny gives the extended Indo-European root as *bhudh-m(e)n with a *bhudh-no- variant, from which the Latin fundus, as in fundament, is derived. The original meaning of English north, from Indo-European *ner- "under", indicates an original sense of "lowlands" for "bottomlands". On the other hand, by "north" the classical authors usually meant "outermost", as the northern lands were outermost to them.
Which meaning prevailed is a distinction that may be too precise to determine, especially as European cultures tended to assimilate and exchange cultural elements.
The third possibility is that botten is a mistranslation of pohja in pohjanmaa, as pohja in Finnish means both north and bottom. The common translation for Pohjanlahti is "the bay in the north," which makes sense. "The bay of the bottom" doesn't make sense, but could have been translated so by a Swedish speaking person who wasn't well versed in Finnish. These types of translation errors are common in Finland, so the explanation seems reasonable. However, whether Pohjanmaa was translated to botten or vice versa is a question for history, archaeology, and politics, and relates to who settled and named the region first.
Into the gulf flow a number of rivers from both sides; consequently, a salinity gradient exists from north to south. In the south the water is the normal brackish water of the Baltic Sea, but in the north, in the Bothnian Bay, the salinity is so low that one can no longer taste the salt in the water and many freshwater fish thrive in it. Being nearly fresh, the gulf is frozen over five months every year. The icing of the Baltic sea begins and ends there.