Prepare vinaigrette just before use. Through the feed opening of a running blender, add 1 at a time: champagne vinegar, lemon juice, chopped mint, salt and pepper. Leaving the blender running, add the oil in a slow stream. Adjust seasoning, to taste, with salt and pepper.
Toss together arugula and onion. Fold in watermelon. Gently stir in enough of the vinaigrette to coat. Top with feta cheese. Serve with any extra dressing on the side.