Maybe because it's the only borough that's not an island -- it's part of the mainland. It's just a guess.

"The Bronx was called Rananchqua by the native Siwanoy band of Lenape (the Delawares to Europeans), while other Native Americans knew the Bronx as Keskeskeck. It was divided by the Aquahung River. Jonas Bronck (ca. 1600–1649), a swedish sea-captain working for the Dutch, entering New Netherland in 1639, became the first recorded European settler in the area. He leased land from the Dutch West India Company on the neck of the mainland immediately north of the Dutch settlement in Harlem (on Manhattan island), and bought additional tracts from the local tribes. He eventually accumulated 500 acres (about 2 square km, or 3/4 of a square mile) between the Harlem River and the Aquahung, which became known as Bronck's River, or The Bronx. Dutch and English settlers referred to the area as Bronck's Land. The American poet William Bronk is a descendant of Pieter Bronck, either Jonas Bronck's son or younger brother. The Bronx is referred to, both legally, and colloquially, with a definite article, as The Bronx. (The County of Bronx, unlike the coextensive Borough of the Bronx, does not place the immediately before Bronx in formal references, nor does the United States Postal Service in its database of Bronx addresses.) The name for this region, apparently after the Bronx River, first appeared in the Annexed District of the Bronx created in 1874 out of part of Westchester County and was continued in the Borough of the Bronx, which included a larger annexation from Westchester County in 1898. The use of the definite article is attributed to the style of referring to rivers. Another explanation for the use of the definite article in the borough's name is that the original form of the name was a possessive or collective one referring to the family, as in visiting The Broncks, The Bronck's or The Broncks'." Source and further information: http://en.wikipedia.org/wiki/The_Bronx

It's just like the Hague, the Netherlands, the North Pole, the Ozarks, the Philippines, El Salvador, for instance. Some places are just named that way.