In a large, non-reactive bowl, mix onion, garlic, lemon juice, olive oil, white wine, Worcestershire sauce, hot pepper sauce, pepper, celery salt, and salt. Place Cornish game hens in the bowl, coating evenly with the marinade mixture. Cover, and marinate in the refrigerator at least 4 hours.
Preheat an outdoor grill for medium heat, and lightly oil grate.
Cook hens about 1 hour on the prepared grill, brushing occasionally with remaining marinade, until no longer pink and juices run clear.