Combine the sugar and 1 cup water in a small saucepan. Bring to boil over high heat and stir to dissolve sugar. Allow to cool to room temperature, then cover and refrigerate until chilled.
Juice the lemons with a KitchenAid® Citrus Juicer Attachment. In pitcher, stir together the chilled syrup, lemon juice, and remaining 4 cups water. Garnish with lemon slices and serve chilled over ice.
For the basil simple syrup:
Combine the sugar, water, and basil leaves in a small saucepan. Bring to a simmer over medium heat, stirring occasionally, until the sugar dissolves. Remove from heat and let sit for 30 minutes at room temperature. Strain and chill.
For the ginger simple syrup:
Cut the ginger into thin, round slices, then slice it into small pieces. Place the ginger, water, and sugar in a medium saucepan. Bring to a boil over medium-high heat, then reduce the heat to a simmer. Allow the mixture to simmer for 30 minutes, then remove from heat and let cool. Strain the syrup through a fine mesh strainer into a small, sealed container.
Note: If peeled prior to simmering, the leftover cooked ginger will be candied and can be used as a mix-in for cookies, cakes, or even ice cream.
For the blackberry simple syrup:
Bring the blackberries, sugar, and water to a gentle boil over medium-high heat in a medium saucepan. Reduce the heat and simmer until the berries are falling apart and the sauce has thickened, 20 to 25 minutes. Strain the syrup through a fine mesh sieve. Refrigerate at least 2 hours or overnight. The syrup will thicken as it chills.