For the ice cubes: Cut each lime slice in quarters and place into 16 ice cube molds. Combine the water and lime juice and pour over the lime slices. Freeze for at least 4 hours until frozen.
For the syrup: In a small saucepan, combine the water, sugar, and ginger over medium heat. Bring to a boil, reduce heat, and simmer for 5 minutes, stirring occasionally, until the sugar has dissolved. Remove the pan from the heat and allow the mixture to cool, about 20 minutes. Cover the pan and refrigerate for 4 hours.
Chill 4 martini glasses in the freezer. Fill a cocktail shaker with ice. Add the vodka, ginger syrup, and sparkling water. Shake for 10 seconds and strain the mixture into the prepared glasses. Place 1 to 2 lime-flavored ice cubes in each glass and serve.