Byzantine Catholics are not known for brevity. Our services do take some time. Our best-known prayer for the deceased, however, is also our shortest prayer, a mere two words: "Eternal Memory."
Folks who don't know better may instinctively say "memories" (with a nod perhaps to the famous Barbara Streisand song) but that's not quite right. Our memories of our departed family and friends are for us to treasure and pass on. The good deeds a person has performed in this life may live on after him or her. The way they share their resources with others, during life and even after death, may continue to do good for others and keep their memories alive. But none of that is what the Church means when we pray "Eternal memory."
When the Old Testament describes God intervening in the lives of His people to save them, we often see the expression "God remembered..." God remembered Noah (Genesis 30:22) and brought the Flood to an end. God remembered His covenant with Abraham and began the Exodus (Exodus 2:24; 6:5). The Lord remembered Rachel and Hannah and granted the sons who grew to become important in the history of Israel (Exodus 3:22; 1 Samuel 1:19). The Psalms (25:7) teach us to pray to the Lord, "Remember me!"
The prayer "Eternal Memory" entrusts the departed soul to the care of God and asks that God's plan of salvation be fully realized for him or her. The Lord knows each of us from the moment of our conception (Isaiah 44:2; Jeremiah 1:15; Psalm 139:16) has created every one of us to share His life and love forever. Jesus, the Good Shepherd, reassures us that He knows each of us by name and calls each of us to follow Him to the Kingdom of Heaven (John 10:14).
Of course, not every choice people make in their lifetime brings them closer to that heavenly goal. And so, the Church repeats its other brief prayer, "Lord have mercy," again and again for every departed soul. Our repentance is required to open us up to the grace and mercy God desires to pour out on each of us.
We can take an example and draw courage from what happened on Good Friday. The convicted criminal crucified along with Jesus was promised, "Even today you will be with me in Paradise." All the Good Thief had asked was, "Remember me when You come into Your kingdom? (Luke 23:42-43). This is what we ask for our beloved departed when we pray "Eternal memory."