Not all prayers are equal.
There is quite a range of activity that people would qualify as a prayer. Everything from a simple grace before a meal, to silent prayers hoping for safety, to a 6 hour long meditation session on top of a mountain.
The controlled breathing and reciting of mantras found in a meditative prayer can definitely alter a person's vital metrics and brain environment. This in turn affects how they think or focus their thoughts.
The "oh shit" prayers, like when someone is in trouble, might have a calming effect with the placebo effect, like was mentioned. This calming might in fact help the person out of their situation by preventing panic from setting in. Of course the person's belief will not allow them to attribute any benefits of the prayer to anything except the deity it was directed at.
The "I want something" prayers, are just people fantasizing about greener pastures in a religious context.
Saying grace is a formality, like saying thank you.
Some of these prayers might drastically alter a persons ability to think, and others will have about as much impact as blinking.