I think the best way I ever heard the question of prayer answered was during a church camp during my youth.
Think about the pitcher and the batter both praying as they face off. The batter is praying for a home-run and the pitcher is praying for a strike-out. One prayer can be answered; even if one did not believe in prayer, only one can have their wish granted. Sometimes, God chooses to answer one prayer over another based upon things we simply cannot understand.
Unfortunately, the answering of prayer is something that believers will not come to understand until they sit at the feet of God.
The evidence that prayer works is one which cannot be studied. It is based solely upon personal experience. So is there evidence? The only evidence comes from the stories of believers. Unfortunately, this is subjective and not scientifically and factually based.