Combine rhubarb pieces, ginger pieces, and lemon juice in a food processor and process until very well chopped, about 30 seconds.
Strain rhubarb mixture through a fine-mesh strainer or cheesecloth, squeezing to remove all liquid from the pulp. You should end up with about ½ cup liquid. Discard pulp. Pour liquid into a 1 quart container, such as a mason jar.
Add the sugar, and fill to 1 quart with cold water. Cap mason jar and shake vigorously to mix until sugar is dissolved (if using another type of container, give everything a vigorous stir instead). Pour into glasses over ice and garnish with a lemon slice if desired. Store lemonade in a sealed container in the refrigerator.