Place eggs in medium saucepan; cover with cold water. Bring to a boil. Reduce heat; simmer about 15 minutes. Immediately drain; run cold water over eggs to stop cooking. Peel eggs.
In measuring cup or old coffee mug, combine 1/2 cup water and food color. Dip peeled eggs 1 at a time into mixture for several seconds until of desired color. Pat dry with paper towel.
Cut thin slice from wide bottom of each egg so it will stand straight. With small sharp knife, cut off top of each egg about 1/3 of the way down, either straight across or with small slanted cuts to make a sawtooth edge. Carefully remove top portion and yolk; reserve tops. Place yolks in small bowl.
Add mayonnaise and relish to yolks; mash and mix thoroughly with fork. With small spoon, refill large section of egg with yolk mixture, heaping filling. Top each with reserved top portion of egg. Refrigerate 30 minutes or until chilled.