You can try using whatever "medium" the pen uses. Sharpie permanent markers use alcohol as the medium, so rubbing alcohol will dissolve Sharpie ink on most surfaces. The marker may say what medium it uses or maybe you can find it on the interwebs. Rubbing alcohol is worth trying on just about any ink.
But you may be screwed. Blue pigment is often created from natural pigments which tend to stain things, including plastic mats. If that is the case, there is not a whole lot you can do besides tossing the marker, never using that color/brand combo again, and getting a new mat.
Going forward, don't leave ink on your mats any longer than necessary, no matter what kind they are - even dry erase becomes difficult to remove after a while. Stick to less vibrant colors. The really bright reds and deep blues are often using natural pigments (they stain stuff, permanently). Test every marker on a corner or flip-side before using it to see how easily it cleans up. Don't use anything that wants to stay on the mat.
Several years back I used different colors on my erase mats too. I quickly learned that it was much harder to eliminate non-black colors. Eventually I think the fainter stains will disappear as you wash the mat. Best thing to do is to stick with black and not make your lines real heavy. Hope it erases better for you.