It's based on the story of the Niland Brothers during WWII.
Private Ryan in Saving Private Ryan never existed as he was portrayed in the movie. His story, however, was based on an actual event that happened to a member of Easy Company from the 506th regiment of the 101st Airborne. (This is the same company featured in the book and mini series Band of Brothers)
According to Stephen Ambrose, author of Band of Brothers, a few weeks after D-Day, Easy Company went into defensive positions south of the French city of Carentan. One day, one of the company's members, a man named Fritz Niland, came down the line to say goodbye to his buddies because he was flying home.
The story he related to his friends was tragic. Niland had had a brother named Bob in the 82nd Airborne, a division that also parachuted into Normandy with the 101st. Upon arrival at the 82nd, Niland learned that Bob had been killed on D-Day while manning a machine gun.
So Fritz Niland went to the 4th Infantry Division to see his other brother and tell him of Bob's death. Upon arrival there, he discovered that that brother too had been killed on D-Day after landing on Utah Beach.
By the time he got back to Easy Company, a priest was looking for him to tell him that his third brother, a pilot in the Chinese-Burma-India theatre, had been shot down in that same week.
Fritz's mother had received all three telegrams on the same day.
Thus, the Army decided, under their 'Sole Survivor' policy to remove him from the combat zone as soon as possible. There was no search and rescue mission to find Private Niland as portrayed in the movie.
There is some disagreement about how the story actually played out. Members of Easy Company seem to remember it as Ambrose relates it and how I explained it above, but the priest who told Niland of his third brother's death relates the story in his own memoir and says that it was he who told Niland of all three of his brother's deaths, and that Niland's mother was not a widow and did not receive all telegrams on the same day.
Nevertheless, Hanks and Spielberg were fans of the memoir and liked the story enough to turn it into a fictional movie, and most of what happens is completely made up. However, in the movie, Private Ryan is still a member of the 101st Airborne. Hanks and Spielberg later took the novel "Band of Brothers" and made it into a 12-part mini series.
And, to answer the original question, Private Niland was from Tonawanda, New York.