This great question comes from Mark Gervais:
One thing that has been frustrating me with Illustrator is the centerpoint when using the Rotate tool. Since I have the Rotate tool selected there is a ceterpoint defined by the cross hairs in the middle of a selected shape. Often times I'll hold down the Command key (to switch to the Selection Tool) and click on a shape and drag a copy of it by holding the option key. When I release the Command key and try to rotate my newly duplicated shape the center point is still back over in the middle of the original shape. Now rotating the new shape moves it in an arc as opposed to rotating it from it's own center point. I guess my question is... Is there a way to make sure the center point of a shape is always locked to the new shape when drag-duplicating? I don't know how many times this has bugged the heck out of me but it's pretty much on a daily basis. arrrg.
Not every question that is asked at the Real World Illustrator Blog has an answer. But then again, much of life is about finding another way that works. In this case, I can say that I share your frustration. Since you mention that you use the Command and Option keys to both switch to the Selection tool and drag a copy of your object, I also assume that you're using the "R" keyboard shortcut to switch to the Rotate tool. So here are two "solutions":
1. Don't use the Rotate tool. Instead, use the Free Transform tool (keyboard shortcut "E"). The Free Transform tool doesn't have a user-definable origin point and always uses the center of an object. So your keyboard shortcuts and workflow are the same, but you're using the Free Transform tool instead of the Rotate tool. Give it a try.
2. After you've dragged a copy of your object, press the Command (Ctrl on Windows) key to switch to the selection tool and click outside the object to deselect it -- then, while still holding the Command key, click the object to select it again. The origin point for the Rotate tool will be reset to the center of the new object.