The first thing to grasp is that, even though those offering prayers often think of the activity as having influence on a supernatural agent, the real way prayer works is not mainly via this assumption but by an influence of the activity on the spirit of the person praying. Or, if the term spirit sounds supernatural to you, on the mind of the person praying.
This is the way I believe prayer works as well, the effects you describe are precisely what placebo effect is. I understand that this works, even if I'm not sure of the mechanism. Daniel Dennet provides an interesting explanation for the placebo effect, namely how it would have been beneficial to have been an evolved phenomenon.
It seems to be important to you to give parsimonious explanations, but what you miss is that prayer is not science, nor even an instrumental activity like generating electricity.
I'm not mistakenly categorizing prayer in a way that I shouldn't. I understand it is more meditative and soothing, and to explain it in the moment would be ridiculous. But this is not the moment. This is a forum where we discuss such things at arm's length. If I get a phone call about some family emergency, I won't come on to Booktalk and critically analyze it. I will turn off the computer and close my eyes and meditate, to get emotionally centered. The atheistic version of prayer, if you will.
I understand this is the not place to pray. But it is certainly the place to examine prayer. If prayer has always been one of those phenomena that you refuse to examine, I would wonder why. Are you afraid that if you examine it too closely, that even the placebo effect will disappear?
So, to return to the immediately preceding discussion, the fact that thoughts are my thoughts does not preclude them from being the thoughts of God as well.
Do you believe your thoughts are sometimes the thoughts of god? If answers are able to coalesce from your own brain without any outside assistance, then what reason do you have to believe there is another intelligent agent influencing your thoughts? It's an extremely untenable idea, and I can't find any way to make sense of it. It may be comforting to believe, but it is simply nonsensical. The human mind is incredible and easily capable of an infinite number of thoughts(not the total lump sum, but the possible recombination of ideas.)
It makes all the sense in the world to see the god of prayer as the placebo effect, and I've seen no reason nor evidence that leads to a different conclusion.