In intercessory prayer, there are three actors: Me, God, and Other. I think the value of intercessory prayer for Me is easy enough to consider. We need to train our spiritual/emotional functions just like our physical and intellectual functions. There is value in exercise, reading, and prayer. I have not fully developed the idea here, but I think I have it.
The next step is reaching God. To relate this idea I believe I need to turn to experience. I need to remember those times when I looked to what was beyond myself when faced when a problem I could not address. For example, on September 11 I was in New York State and was able to donate blood, but I also turned to God and prayed for healing. I needed to do that in a way that is hard to explain.
Last weekend I helped people who were legal permanent residents take the next step toward citizenship. There were many tragic but stories, but in one particular case, after the woman had told me everything and after I gave her all of the advice I had to give and after knowing that her case was a lost one, we prayed. Again, we needed to bring God into the picture.
Again, underdeveloped, and I know less convincing to the non-believer, but I am confident that it is meaningful to say God hears our prayers.
That leaves the tough part.