T.R.
The EXACT same thing just happened to me a few weeks ago on our new wood table that was less than one week old (with a Sharpie marker, too). I couldn't find anything to take it off so I tried Soft Scrub. While it did take the marker off of the wood, it did dull the wood where it was applied. I don't think anyone can notice it as it's really only noticable in the sunlight if you are looking for it. I also got to the stain within a few minutes...not sure if that would make a difference or not.
I can't believe your cousin isn't offering to help you fix this. It was HER kids that SHE wasn't watching!! If it were me in your cousin's place, I would offer to replace the furniture even if it meant having to pay for it over time on credit.