Combine lime zest and 2 tablespoons sugar in a food processor. Pulse until combined. Set aside.
Place lime juice, strawberries, rum, 2 to 4 tablespoons sugar, depending on the sweetness of your strawberries, and ice cubes in a blender. Blend until smooth.
Rub the rims of four glasses with a lime wedge, dip rims in lime sugar, and fill glasses with daiquiri.