What is the power of a prayer? I think it may have something to do with what's Step 3 of Alcoholics Anonymous' 12 Steps (though I was never an alcoholic, I use this out of relevance):
[We] made a decision to turn our will and our lives over to the care of God as we understood Him.
There's something very comforting in that. Prayer is giving oneself to someone, something greater, because the situation we are in is beyond our means to understand or correct or both. I remember that even before I was a Christian, I prayed. We who are in the world are very small, if you come to think of it. Sometimes, we need a hand to pull us out of depths that are beyond the limit of our fingers' reach.