Source: Dictionary of American Family Names ©2013, Oxford University Press
You can use census records and voter lists to see where families with the Lalli surname lived over the years.
While the questions in census records vary from place to place, and year to year, you can generally find information like name of household members, ages, birthplaces, residences, occupations, etc.
from the New York Passengers List
The information for this chart came from the U.S. Immigration Collection at Ancestry. You can find out where the majority of the Lalli families were living before they immigrated to the U.S and learn where to focus your search for foreign records.
Immigration records can tell you an ancestor's name, ship name, port of departure, port of arrival, and destination.