La Liberté éclairant le monde
(Liberty Enlightening the World) is a gift from the people of France to the U.S.A. to recognize what the French saw as their co-developers of national liberty.
The statue is of a robed female figure representing Libertas, the Roman goddess of freedom, who bears a torch and a tabula ansata (a tablet evoking the law) upon which is inscribed the date of the American Declaration of Independence, July 4, 1776. A broken chain lies at her feet. [Wikipedia]
It didn't have much to do with immigration. The message to the world was that America and France had liberated their own countries and you should admire this statue, then be inspired to go home and liberate your own country.
Not surprisingly, the French never felt oppressed enough in their nice country to immigrate to America in vast numbers. So, Franco Americans tend to be a rather random collection of worthies who filtered in in small number. The Wikipedia on "French-American" article lists these representative Franco-Americans.
Not bad, if a pretty random set of Americans.