Make sauce: Combine barbecue sauce, ketchup, onion, lemon juice and Tabasco sauce in a small bowl and whisk until blended. Serve at room temperature or chill.
Make kebabs: Preheat broiler or prepare a charcoal fire and let burn to a gray ash. Toss chicken with Worcestershire sauce, lemon juice and olive oil in a bowl until well mixed. Set aside to let chicken marinate at room temperature for at least 15 minutes. (The chicken can marinate, covered, in the refrigerator for up to 24 hours.)
Thread chicken cubes onto 8 long metal skewers, alternating with onion pieces. Brush each skewer with olive oil and season with salt and pepper
Set broiling pan or grill about 6 inches from heat source. Broil or grill kebabs, turning often, until chicken is lightly browned and cooked through and onions are lightly charred on ends, about 10 minutes. Serve kebabs hot with barbecue dipping sauce.