NY Times: The short answer is that the borough was named for the Bronx River, and the name was shortened to the Bronx for convenience. If Manhattan had been named for its main river, we would probably be calling it the Hudson.
Jonas Jonassen Bronck was a Swedish-born sea captain who arrived from Amsterdam in 1639. “When he died in 1643, at the age of 43, the only thing that was named for him that kept his name through the centuries was Bronck’s River,” Professor Lloyd Ultan, the Bronx borough historian, said. “It was much easier to abbreviate it as Bronx.”
In 1898, when the city set up the borough system, it was decided that the annexed areas north of Manhattan should also become a borough, but it had never had one name before, the professor said. People in Manhattan called it the Annexed District; people in the district called it the North Side.
In the middle of the territory was the Bronx River, and so the borough was named.