Why is St. Anthony invoked for finding lost items?
The short answer
Because of a popular story: a brother friar stole a commentary Anthony had written; Anthony prayed, and the thief, frightened by a sudden storm he took as God's wrath, returned it. Whether the story is true or a pious legend, we don't know for certain.
The story
A fellow friar is said to have stolen a commentary St. Anthony had written. Anthony prayed for its return. The friar was caught in a sudden storm that so frightened him he took it as God's wrath for his sin, and he ran back to return the manuscript.
Since then
So the story goes, St. Anthony has had a special gift for recovering lost things.
True or legend?
Whether it is true or a pious legend is not known for certain.