Heat grill. In food processor, pulse ginger and the garlic until finely chopped. Set aside half of mixture. Gently loosen skin from breast and thigh of each hen half. Insert about 1/2 teaspoon of remaining ginger-garlic mixture under skin of each hen half.
In small saucepan, combine remaining ginger-garlic mixture, hoisin sauce, chili sauce and preserves. Over medium-high heat, cook, stirring, until preserves are melted.
Grill hens, skin side down, 8 minutes. Turn over. Baste with some of sauce. Grill hens over indirect heat, with grill covered, until instant-read thermometer inserted in breast registers 170 degrees F, 30 to 35 minutes, basting occasionally with some of sauce. Serve with coleslaw or potato salad, if desired. Boil any extra sauce 3 minutes before serving.