Combine sugar and water in a saucepan over medium heat, stirring occasionally. Bring to a boil, and then reduce to simmer until sugar dissolves, about 5 minutes.
Remove saucepan from heat and allow to cool, about 30 minutes. (You can make this syrup up to 1 month in advance. Store in an airtight container in the fridge until ready to use.)
Puree strawberries and mint leaves together in blender or food processor.
Combine all the simple syrup, strawberry-mint puree, lemon juice, water, and vodka in a 2.5-quart pitcher. Stir to combine.
Fill rocks glasses with ice and pour in the lemonade. Garnish with a lemon slice or wedge and some more fresh mint, if desired.
Notes
Fresh lemon juice tastes best in this drink, but you can substitute bottled juice if needed.