The origin of the Bridgeport name "is shrouded in myth, purportedly owing to a low bridge spanning one of the waterways which forced a transfer of cargo from larger to smaller vessels." How it got to that point, though, isn't that much of a mystery. Bridgeport, like many other neighborhoods in the city, went by various names before settling on what we all know it as today. Dating back to the early 1800s, the first American settlement in the area "were the farmer dwellings connected with the Charles Lee and Russell farm." The farm sprouted up shortly after the establishment of Fort Dearborn, and even though Mr. Lee didn't live on the property, the land was known as "Lee's Place" by locals.
The next name that the area became known by was Hardscrabble, a change that happened around 1818 when the area was mainly serving as a fur trading outpost. At it's height, Hardscrabble consisted of "several cabins, a trading post, and a lodging house." In 1835 Hardscrabble gave way to Canalport, a name that was given to the nearby settlements when the area around the canal was platted. Canalport didn't last long, and the name was changed to Bridgeport soon after.
(So where does "Cabbage Patch" come from? This Chicago Tribune article about Bridgeport included this passage: "For a time Bridgeport was known as the Cabbage Patch. That was when people grew cabbages in their yards." I wasn't able to find what years the neighborhood was called this.)