Three suggestions for your research—both to garner specific information about individual family members and context for their lives in Paterson:
1) Most of the immigrants who lived in or close by to Paterson were industrial workers, so finding out in which industries family members worked, their employer names, and location of their workplaces would garner you some important information (WWI draft cards would be a good source for this information). The central industry in Paterson was silk (hence Paterson’s moniker “Silk City”) at least into the1920s, so if the family was living in Paterson up to then it’s a good bet that some of its members worked in the silk mills. If the family lived in Paterson before 1913 they may have been involved in the famous silk strike of 1913 about which there's a great deal of information
2) Directly across the city line from Paterson is the town of Haledon. Many who worked in the mills and other industries in Paterson lived in Haledon (and a lot of the rallies during the 1913 silk strike took place there,too), particularly Italian immigrants—many of them skilled.. There’s a truly excellent book on Haledon and its immigrants that has a lot of information about Italian immigrants to the area (not all came from Southern Italy) that’s part of Arcadia Publishing’s Images of America series: Around Haledon: Immigration and Labor. A blurb for the book describes it this way:
By 1908, when Haledon became independent from Manchester Township, thousands of southern and eastern European immigrants settled in the borough and its surrounding area. Immigrants found work in textile mills, machine shops, and other industries located in proximity to the city of Paterson and the Passaic River and its mighty Great Falls. Land promoters spurred home building in Haledon, a streetcar suburb. In 1913, nearly 25,000 workers went on strike, demanding an eight-hour workday. During the six-month strike, Haledon became the workers' haven for free speech and assembly as they demanded safer workplaces, a living wage, and an end to child labor. Archival photographs, documents, and postcards from 1890 to 1930 share the story of workers and immigrants who fought for the workplace benefits widely enjoyed by Americans today.
3) One of the book’s authors, Angelica Santomauro, is the current director of the American Labor Museum, so you might want to contact her to ask about local and national sources and Italian-American genealogical organizations. (https://labormuseum.net/?p=staff-listing&team_id=67)
Teaneck, New Jersey