For marinade, in a screw-top jar, combine cooking oil, lime juice, water, onion, tarragon, salt, ginger, pepper sauce, garlic powder, and black pepper. Cover and shake well.
Place chicken in a plastic bag set in a shallow dish. Pour marinade over chicken. Close bag. Marinate in the refrigerator for at least 1 hour or up to 2 hours, turning bag occasionally.
Drain the chicken; reserve marinade. Place chicken on the rack of an uncovered grill directly over medium coals.* Grill for 12 to 15 minutes or until chicken is no longer pink, turning once and brushing with marinade halfway through grilling. Discard marinade. Makes 4 servings.
For medium heat, you should be able to hold your hand over heat at the height of the food for 4 seconds.