1. Combine half each of watermelon, mint, juice, and cucumber in a blender or food processor; process until smooth. Line a fine sieve with 4 layers of cheesecloth, allowing cheesecloth to extend over edges; strain watermelon mixture through prepared sieve over a bowl, reserving juice mixture. Gather the edges of the cheesecloth together. Holding cheesecloth over sieve, squeeze to release the remaining juice mixture. Discard solids. Repeat the procedure with the remaining watermelon, mint, juice, and cucumber.
2. Combine 2 1/2 cups juice mixture, tonic water, and gin, stirring well to combine. (Reserve any remaining juice mixture for another use.) Serve over ice.
Hendrick's, a small-batch Scottish gin that's infused with the flavor of cucumber and rose petals, is ideal for this recipe. For even better results, try premium tonic water, such as Fever-Tree or Q Tonic. Serve any remaining juice mixture (which you can make a day ahead) over ice for guests who prefer a nonalcoholic drink.