You may have valuable clues in old family papers. If your immigrant ancestors arrived after 1880, naturalization papers often include a specific town of origin. For earlier immigrants, the most useful information is the immigration date. That date can be obtained from some census records and naturalization records. Death certificates sometimes include the actual place of birth, but most do not.
Obituaries are usually an excellent source of biographical information about immigrants.
In addition to names and death dates, you will discover the names of surviving family members, their church affiliations, name of spouse, occupation, burial places and sometimes the birthplace. And, if your ancestor's obituary is not helpful, check those for all of his or her siblings. Obituaries were usually published in local and church newspapers, and are most common after the mid-1800s. You are most likely to find obituaries of immigrants who lived in small towns and rural areas rather than in large cities. However, many ethnic groups are compiling and publishing these records.