It's a simple request, one we may make or hear made a dozen times a day: "Pray for me." There is a need, a desire, a hurt, a blessing. Something needs to be addressed, something the petitioner can't do on his or her own. They request prayer, hoping the one they ask will take up their cause, approach the throne of God, and make intercession for them.
Do we? Or when we tell someone "I'll pray for you," does it simply . . . never happen?
I'll be the first to admit I used to be pretty bad at praying for people. I'd say I would, then go on about my day and never think about it again. It's not that I was being deliberately malicious, mentally rubbing my hands together and thinking, God will never hear this, muwahaha! It was just, as a human being, life happened. My next designated prayer time may not have been until hours later, and so their request would be replaced in my mind by thoughts of work to be done, meals to eat, errands to run, and other people demanding my time.
I doubt my story is unique. Most if not all of us have done that at some point. But prayer is important. Prayer is a direct connection to the Almighty, and the person thought the occasion important enough to ask you to utilize it. It wasn't a trivial, forgettable concern for them; it was something requiring divine intervention.
When someone asks you to pray for them, do it. Right then. Even if I don't pray with them, as soon as they hang up or walk away, I pray for them. Maybe that sort of immediate prayer will work for you, too. Or maybe you could carry around a notepad just for prayer requests you've received over the course of the day and pray for them all at once that evening. Do whatever will work for you.
Remember, Christ died for them, too. He wants us to pray for each other and bear one another's burdens. When we do this, we truly become united into a single body of Christ who loves each other and takes that love to a lost and dying world. The world needs to hear about Jesus -- because that's good news!