Grate 2 tablespoons each of lemon and orange zest. Squeeze 1/3 cup lemon juice from 3 of the lemons. Set aside.
In a food processor, pulse the ginger, honey, and sugar until just combined. Add the orange and lemon juices and process. Transfer to a large lidded container. Add the orange and lemon zests and 2 cups of boiling water and stir until the sugar dissolves. Cool to room temperature. Cover and refrigerate for at least 24 hours and up to 5 days.
To serve, strain the ginger-beer base into a pitcher. Thinly slice the remaining lemon and orange to form circles. Add the slices and mineral water to the pitcher and stir. Serve over ice.