Barbie dolls are made up of soft plastics, and while permanent marker has been known to create irreversible damage, there are some materials, such as rubbing alcohol, that have been proven to erase the marker from select surfaces. Removing permanent marker stains should be done on a trial-and-error basis.
Rubbing alcohol and nail polish remover have been known to remove stains, mostly because of their acetone content. Mr. Clean Magic Eraser and Goof Off are two known brands of cleaning solutions that are also used to clean marker stains. Other lesser known stain removers are insect repellant, baby wipes, anti-bacterial hand gel and ammonia with water.