Looks for the Pivot Center drop-down menu here:
Left-Click and in the drop-down menu select 3D Cursor:
Now the object should rotate around the 3D Cursor (that target thingy) and will animate correctly as long as you key-frame using Location and Rotation, This is the fourth key-framing option down when you use the key-frame menu (I-Key):
You need to use this option now rather then just key-framing the Rotation because if you are manipulating an object around the 3D Cursor your are also adjusting its location, this shouldn’t be a problem if you are already keyframing using the Loc/Rot/Scale option, but if you do need to add the key-framing for Rotation don’t worry you shouldn’t lose any animation data when adding the needed key-frames.
Note: If you press period-key (.) it will also change the pivot point around the 3D Cursor, additionally the comma-key (,) changes to Bounding Box Center, CTRL-, changes to the Median Point (This is the default), CTRL-. changes the pivot point to Individual Origins, Alt-. Changes it to Active Element and Alt-, will activate manipulation of only origin points which can be helpfull if you want to move an object around the cursor for example and not actually change the local rotation of the object itself. All of these options can be activated through the same drop-down menu, with the exception of activating manipulation of only origin points, this is the button found immediately to the right of the Pivot Center drop-down menu.