My family has made what we call "Red Beet Eggs" long before I was born. Our recipe uses the following ratios:
1 cup sugar
1/2 cup cider vinegar
1/2 cup beet juice
We use canned red beets, either sliced or whole, and drain the juice into a 1/2 cup measuring cup and if not quite enough we add water to make a 1/2 cup measure.
After the eggs are hard-boiled and peeled we place them in an empty pickle jar. We heat the sugar/cider/ beet juice until the sugar dissolves then pour it over the eggs and last of all, pour the beets on top and refrigerate. I like to let it "do its thing" for at least 3 days before serving but they can go much longer.
I usually do at least 18 eggs, use 2 cans of beets and double the sugar/cider/vinegar mixture.