In a large pitcher, combine the sliced cucumber, mint leaves, ginger, and lime juice. Use a muddler or the back of a wooden spoon to gently press and twist the ingredients for about 30 seconds to release their flavors.
Pour the cold water over the muddled mixture. Stir well, then cover and refrigerate for at least 10 minutes to let the flavors infuse.
If you prefer a sweeter drink, stir in the honey or agave until dissolved. Taste and adjust.
Fill four tall glasses with ice cubes. Strain the infused water into the glasses, dividing evenly. If you like, drop a few of the muddled cucumber slices and mint leaves into each glass for garnish.
Top each glass with sparkling water (about 1/4 cup per glass) and give a quick stir. Serve immediately while it's cold and fizzy.