St. Anthony was named patron saint of finding lost objects stems from his time as a Franciscan friar. He was a famous orator and also a teacher. One day, a student in his abbey decided to quit; he no longer wanted to become a priest and basically went AWOL. However, in the process of leaving, the bitter ex-priest stole St. Anthony's personal psalter, containing all of his personal notes for sermons. Devastated, Anthony prayed for his psalter's return, wanting a lifetime's worth of notes back. In response to his prayer, God changed the ex-priest's mind and had the priest give the psalter back. The townspeople were so impressed that they began praying through Anthony to find lost objects after his death; after all, if God returned such an important item to him, why couldn't he help ordinary people?
An alternate version of this story is that a demon appeared in the path of the ex-priest and forced him to return the psalter. However, this tale was discredited by the Catholic Church. After all, why would an inherently evil creature come to the aid of a good priest?
Thanks to americancatholic.org information on St. Anthony.