I, too, am torn on how I feel about the name. I do not like just calling it "The Black Bottom," but I think having it called something like "The Heart of the Black Bottom" gives it something more and makes it seem warmer.
The universities in the surrounding area have been trying to make the area a better and safer place both for the residents in the neighborhood and their students. They have been trying to redevelop the area to make it a more desirable place to live.
It seems as though people from the neighborhood are ok with the area being named the Black Bottom and even embrace it. People from the old neighborhood declared the last Saturday in August, Black Bottom Day and celebrate the day in Fairmount Park.