First, I'm grateful your church has this ministry, for nothing is as important as prayer in God's work. (A "prayer chain" is a list of persons who are notified by telephone or e-mail, and are asked to pray for people who are sick or facing other needs.)
Admittedly, it's easier to pray for someone we know rather than someone we don't know. But has it ever occurred to you that those you know the least may need your prayers the most? They may, in fact, have very few friends or others who will pray for them. If you were in their situation, you would be very grateful for every person who would pray for you.
Don't let the fact that you don't know someone keep you from praying for them. I have often said that prayer is the foundation of our own evangelistic ministry, and thousands of small prayer groups may be organized in a city in advance of an evangelistic mission. Yes, they pray for their friends but they also pray for people they don't know, that they will come to Christ.
The Bible says, "The prayer of a righteous man (or woman) is powerful and effective" (James 5:16). No, you may not know the person for whom you are praying but God does, and He is able to meet their needs. And in time you may have the privilege of meeting them and knowing God used you to help them in their time of need.