Nolita = NOrth of LITtle ItAly
Soho = SOuth of HOuston (Street)
DUMBO = Down Under the Manhattan Bridge Overpass!
Leave it to the Real Estate hawkers to rename your neighborhood in order to bring the big-bucks from naive newcomers.
Also, I heard that an old Italian woman was evicted from her apartment in Little Italy in order to turn the building into a museum dedicated to the history of Little Italy. I'm not sure if that is the truth or an urban legend.