Place the butter, honey, rum and lemon juice in a wide skillet. Simmer mixture over medium heat until very lightly thickened. Add the blueberries and cinnamon and cook 1 to 2 minutes more. Add the apricot halves and cook until just heated through, about 1 minute per side. Serve warm with vanilla ice cream.
Recipe Options: Use blackberries instead of blueberries. Try another alcohol in the sauce, such as brandy or orange liqueur. Use 2 Tbsp of brown sugar in the sauce instead honey. Make the sauce a more spice-filled one by also adding a pinch of ground clove and nutmeg.