Alright this is all opinion I have never tried to take ink off of a pen before and am not sure if the ink is just on the finish or in the wood.
First the good news the thicker ballpoint ink might not go all the way through the wood (if it got to the wood). I have a fealing Rollerball or Foutain Ink would have worked like a dye and penetrated all the way through the wood.
Rubbing Alcohol can usually be used to remove ball point ink. If you have a shellac based finish (friction polish) it will also melt the finish, not sure about other finishes.
Worth a try but I have a fealing you probably be giving the customer a deal on a new pen.
Good luck if you find a method that works please post it. I know I have seen a post about a washed pen at least once before, but he wasn't trying to fix it.