Mr. Clean Magic Eraser to the rescue!
It'll take hard work and elbow grease and probably most of an eraser ... but they're like fifty cents, so, worth it.
I was careful around the grid lines when I erased my stain, but I'm honestly not sure if the eraser would pick it up.
Acetone will also remove it, but also will remove the tan dye used to dye the mat and the grid lines ... so you'll go, if careful, from a black line to an off-white line.
The mat at our game house still has a big compass rose on it from where the GM picked up one of my tact-tile dry erase markers and started to do a map before I could yell out: "Nooooooooooooooooooo!". Nobody has gotten around to spending a half hour with a magic eraser.
--fje