In a cocktail shaker, muddle or crush the lime and mint leaves. Add in the raspberry simple syrup and white rum. Add ice, cover, shake, and then serve in a rocks glass. Garnish with a raspberry.
To make the simple syrup: In a saucepan combine the sugar and water. Cook the mixture over medium heat until the sugar dissolves. Remove from the saucepan from the heat and add in the raspberries. Allow the mixture to steep for at least one day. The simple syrup will keep up to 2 weeks in the refrigerator. Add the syrup to the drink when called for above.