For the strawberry-basil puree: To make the simple syrup, heat the sugar and 2 cups water in a saucepan over medium heat, stirring as needed, until the sugar has completely dissolved. Set aside and allow to cool completely.
Add half of the strawberries, lime juice and basil to the bowl of a food processor. Pulse until you reach a consistency that makes you happy. I do 10 to 15 pulses, which leaves a good bit of texture. Transfer to a pitcher or container. Repeat with the other half and add to the same container. Pour in the room temperature simple syrup, mix and chill for at least 30 minutes.
For the strawberry-basil mojitos: Pour 1/2 cup of the strawberry basil puree into each glass. Top with 2 ounces of your favorite light rum, add ice and stir gently to combine. Top off each glass with a splash of soda. Garnish each drink with a stalk of basil and a lime wedge.