In a small saucepan, cook onion and celery in hot oil until tender. Stir in apricots, stuffing mix and ginger; set aside.
Place each chicken breast half between 2 pieces of plastic wrap. Use the flat side of a meat mallet to pound into a rectangle about 1/8 inch thick. Remove the plastic wrap from the chicken.
Divide stuffing mixture among chicken pieces. Fold in sides of chicken; roll up. Secure with wooden toothpicks.
In a small bowl, stir together apricot jam and vinegar.
For a charcoal grill, arrange medium-hot coals around a drip pan. Place chicken on grill rack over drip pan. Cover and grill for 20 to 25 minutes or until chicken is tender and is 170 degrees F. when tested with an instant-read thermometer, turning after 10 minutes and brushing occasionally with the brush-on during the last 5 minutes of grilling time. (For a gas grill, preheat grill. Reduce heat to medium. Adjust for indirect cooking. Grill as above.)
You can use 4 dried apricots, snipped, in place of fresh or canned apricots. For dried apricots, stir in enough apricot jam to moisten stuffing mixture (1 to 2 tablespoons of jam).