Place the red onion in a small bowl and cover with cold water. Let sit at least 5 minutes, then drain.
Place the arugula in a large bowl and drizzle lightly with the lemon vinaigrette. Toss to coat.
Add the cucumber, cherry tomatoes, bell pepper, chickpeas, and drained red onion. Drizzle with a little more dressing then toss gently to combine.
Last, add 3/4 of the feta, and toss very gently. Add salt and pepper to taste. Sprinkle the remaining feta over the top.
Divide between serving bowls and serve with additional dressing as desired.