I do think that prayer, like many rituals, is something that the religious get some real benefits from that are just lost to us heathens. One reason is that many of these rituals are performed communally, as part of a regular meeting or worship. This means there is social reinforcement. But the main one is that the religious context transforms them from something optional and arbitrary into something necessary and grounded. Because the rituals are a duty to our absolute sovereign, there is strong reason to keep them up. You pray every day because you sense you really ought to, and it will be noticed if you don't. In contrast, the belief that daily meditation is beneficial motivates in much the same way as the thought that eating more vegetables or exercising is. Inclination comes and goes and needs to be constantly renewed.
I also think praying can be useful for the person suffering from an illness. You'd think it could backfire. The patient might pray instead of getting all the advanced treatments. Yes, there are people who believe in prayer instead of medicine. But what I've observed (here in the religious heart of Texas) is that people who pray will also go to the ends of the earth to find a medical cure. In fact, I believe there may be a connection, sometimes, between praying and aggressively trying to treat disease. Prayer may sometimes give someone just the boost in hopefulness she needs to believe that further treatment might be helpful. At least, that's how it seemed in the case of a woman I met about 15 years ago. A mother of three young children, she prayed mightily for a cure for her breast cancer, while also seeing out every conceivable cutting edge treatment. (Sadly, she did not survive.)
You'd think maybe if a person believed God answers prayers, then she'd have to think God also causes diseases to begin with. So she'd feel better in a way ("God may help me") but also worse ("God let me get this disease"). Could be--but not necessarily! God is good, you might think. The cancer is not God's doing--but the cure could be.
I couldn't possibly believe in the efficacy of prayer. Come on--what God can cure, he could have prevented in the first place. More overwhelmingly, it's impossible for me to believe in a God who let six million Jews die in the Holocaust, but is now on standby to help individuals with cancer. Please. Still, it would be good in very tangible ways not only to be able to pray, but to believe it was beneficial.