Drew Brees
It seems insane now but there were a lot of doubts about Brees before he moved from the Chargers to New Orleans. Many people even forget that he began his NFL career in San Diego. His numbers with the Chargers were unexceptional. They were happy to swap him for Philip Rivers in one of the weirdest moments in NFL history (via L.A. Times).

Perhaps it was the coaching or the change of scenery but it worked. Brees blossomed into one of the most productive NFL quarterbacks in history. He held many passing records until his retirement and won a Super Bowl ring. Meanwhile, Rivers had a decent career with the Chargers but fell short of greatness.