Pour about 1/2 inch of sugar onto a small, shallow plate. Run a lime wedge around the rims of 8 cocktail glasses, then dip the rims into sugar to coat; set aside.

Squeeze all of the lime wedges into a sturdy glass pitcher. Toss juiced limes into the pitcher along with mint, strawberries, and 1 cup sugar. Crush with a muddler to release strawberry juice and mint oil. Stir in club soda and rum until sugar has dissolved.

Pour into the prepared glasses over ice cubes to serve.