Preheat the oven to 400 degrees F (200 degrees C). Grease a 12 cup muffin pan, or line with paper muffin cups. Place apricots into a small bowl, and pour the boiling water over them. Let stand for 5 minutes.
In a medium bowl stir together the flour, sugar, baking soda and salt. In a separate bowl, whisk together the melted butter, oil, buttermilk and egg. Pour the wet ingredients into the dry ingredients, and stir until just blended. It is okay for the batter to have some lumps. Drain water from apricots, and mix them into the batter. Spoon into the prepared muffin cups.
Bake for 15 minutes in the preheated oven, or until the top springs back when lightly pressed. Cool in the pan over a wire rack.