Pfaffenbarger, Puffenbarger, Puffinbarger, and variants on the same are all "Ellis Island Respellings" of the German name Pfaffenberger. Pfaffenberger simply means "a person from a place called Pfaffenberg," which refers to one mountain, one large hill, and one municipality in Germany, none of which are especially close to each other geographically. Pfaffenberg is a compound of pfaffe, meaning "parson," and berg, meaning "mountain." Pfaffenberg is therefore "parson's mountain."
Ellis Island respellings did not all necessarily take place at Ellis Island itself, and more often these spelling errors - usually actually errors, and not deliberate name changes - turn up much later after an immigrant family's arrival in the United States. The most common source of these errors are new birth certificates created for children born to parents with poor literacy, and the United States Census Bureau, which processes enormous amounts of population data over a short period of time, and has more opportunities for errors to crop up. Considering just how many different forms of Pfaffenberger are found throughout the US, despite the name being very uncommon in the wider population, it is likely that these variants developed at different times and locations, rather than all springing from the same occasion of misspelling, such as might have happened at Ellis Island.