For syrup, in small saucepan combine water, sugar and sliced ginger; bring to boiling; stir to dissolve sugar. Cover and simmer 5 minutes. Remove from heat; cool. Strain; discard ginger.
Divide mint among 6 tall glasses or silver mint julep cups. Add 2 tablespoons ginger syrup to each glass. With a spoon, stir to bruise mint. Divide rum evenly among glasses; fill with ice. Add lemon peel.