1. Combine first 3 ingredients in a small saucepan; bring to a boil. Reduce heat; simmer, uncovered, 15 minutes, stirring occasionally. Strain mixture through a sieve into a bowl; discard solids. Chill ginger syrup 1 hour or until very cold.
2. Fill a martini shaker with ice. Add 1/4 cup chilled ginger syrup, 1/2 cup chilled vodka, and 1 tablespoon lime juice. Shake well; strain into 2 martini glasses. Repeat procedure with remaining ginger syrup, vodka, and lime juice. Garnish glasses with lime wedges and mint leaves, if desired.