Heat gas or charcoal grill. In 1-quart saucepan, cook onion and water over medium heat 3 to 5 minutes, stirring occasionally, until onion is tender. Stir in barbecue sauce, preserves, chile and salt. Cook 1 to 2 minutes, stirring occasionally, until heated through. Reserve 3 tablespoons sauce.
Cover and grill drumsticks over medium-low heat 25 to 30 minutes, turning frequently and brushing with sauce during last 5 minutes, until juice of turkey is clear when thickest part is cut to bone (180°F).
Serve drumsticks with reserved 3 tablespoons sauce.