I need to remove permanent marker from hard plastic. How can I do it?
I use pure Eucalyptus oil. It works wonderfully. Keep in mind that if you are using it on LEGO, it will also remove the original printing (but won't harm the plastic itself), so I wouldn't recommend it in that case. But other than that Eucalyptus oil can remove marker, paint, glue-residue, and all kinds of gunk much better and faster than products like Goo-Gone. I'm an artist and use it on a regular basis. Eucalyptus oil is available in small bottles at health-food stores, larger bottles at pharmacies (that's what I use), and also at some Asian stores (usually the cheapest).
Using something you may already have in your home:
Spraying this over permanent marker allows the marker to be wiped off, and then you can rinse off the residue in warm water. I've used or seen used this remedy on many types of plastic in my life, but I can't recall if I've used it on LEGO pieces before. I think I did use it on a yellow brick some 12+ years ago, but time makes the details unclear.
Assuming it's regular alcohol based permanent markers, 70% Isopropyl alcohol will wipe it off easily. Coat a swab or wet a tissue, don't apply it directly to the brick - it slightly eats away at the material, but not to the degree where you'll shorten the lifespan of the plastic without repeated cleaning or prolonged exposure.
Nail polish remover (or purer acetone if you happen to work in a lab) works really well. You can either soak the piece in it (I like to use a small tupperware container) if it is really covered, or wipe it away with a paper towel wetted with the solution.
This works because the sharpie in is a non-polar molecule and thus will dissolve in solvents like acetone. Rubbing alcohol also works like this, but is markedly less effective.