Thanks to substantial genealogical records, many Amish can trace their lineage back to the 1700's and 1800's, when significant waves of Amish immigration to North America occurred.
One of the most common Amish names is Stoltzfus, found predominantly in Lancaster County, Pennsylvania, and its offshoot settlements. The short-version history of the Stoltzfus name, courtesy of Family Life:
"On Oct. 18, 1766 Nicholas Stoltzfus with his two sons and two daughters landed at Philadelphia from Zweibrucken, Germany. This Nicholas Stoltzfus is believed to be the ancestor of all those bearing the Stoltzfus name among the Amish and Mennonites today.
Nicholas purchased a plantation along the Schuylkill River near Reading, Pennsylvania. This he divided into five farms. The oldest daughter, Barbara, married John Schmucker, oldest son of the immigrant Christian Schmucker.
The Stoltzfuses were hard workers and good managers, and prospered financially. Large families among them was the rule.
Christian Stoltzfus (1748-1832) was born in Germany, the youngest son and fourth child of Nicholas Stoltzfus. He was a farmer and Amish bishop and in his later years lived near Gordonville, Penna."
(Source: Family Life, Yesterdays and Years: Amish and Mennonite Family Names Part 4, Joseph Stoll, March 1969)