How to Make It
Bring 1/2 cup water and sugar to a simmer in a small saucepan over high. Simmer, stirring often, until sugar dissolves, 1 to 2 minutes. Remove from heat; add mint, and stir until submerged. Refrigerate until mixture is completely cool, 1 hour. Pour mint mixture through a fine wire-mesh strainer into a bowl; discard solids. Chill mint syrup until ready to use.
While syrup cools, place watermelon in a blender, and process until smooth, about 20 seconds. Pour through a fine wire-mesh strainer into a large measuring cup, pressing gently to squeeze out juice. Discard solids, and refrigerate 15 minutes. Repeat straining procedure. (You should have about 4 cups juice.)
Stir together mint syrup, watermelon juice, rum, ginger beer, and lime juice in a large pitcher.
Pour evenly into 10 highball glasses filled with ice; garnish each with a mint sprig and small watermelon wedge cut to sit on rim of glass.