On a large plate, toss the onion with the vinegar and a pinch of salt. Spread in a single layer and let sit for 20 minutes to pickle.
In a bowl, combine the tomatoes, olives, 1 tablespoon basil and 2 teaspoons olive oil; season lightly with salt and pepper. In a separate bowl, combine the bocconcini, remaining 1 tablespoon basil, remaining 2 teaspoons oil and a pinch of salt.
Fill 4 martini glasses with the tomato salad, bocconcini and onions.