The trick is to dissolve the ink, but not dissolve the plastic. Acetone tends to be rather aggressive toward plastics (it depends on the type of plastic, of course). I know from experience that acetone will etch Star plastic, as in the process of stripping off stamps with it it drove some of the stamp down into the plastic (and picked up some of the color of the plastic on the towel I was using). Lacquer thinner did roughly the same thing.
Some of the simple alcohols (ethanol, isopropanol) provide a viable alternative that will dissolve sharpie ink, but are far less harmful to plastics. I'd try one of these first. Rubbing alcohol or gas line antifreeze would be good bets. Only move on to less plastic friendly solvents (like acetone, gasoline, etc) if the alcohol doesn't work. Actually I'd stay away from gasoline for safety reasons. While all of the above are flammable, gasoline is the easiest to accidentally ignite and the most difficult to put out once burning.
