the active ingredient in nail varnish is acetone, you might be able to buy it cheaper than in 15 ml bottles of nail varnish - depends how many you need to do.
most things will clean best using the same or similar solvents as were contained in the original, hence the reason the 'write over with another marker' trick works is that the ink contains a solvent to keep the ink fluid in the pen and that rapid dries it as it evaporates out. Adding fresh marker adds more solvent - might take a few goes though.
The Sharpie site also has some Q&A's. this is from it.
'The black ink in the Fine, Twin Tip, Chisel and Super is permanent ink. The principle solvents are alcohols, but they also contain ethylene glycol monobutyl ether. All other Sharpie ink colors are Permchrome ink. For these the principle solvents are also alcohols, but no glycol ethers are used.'
It also suggests AMODEX STAIN REMOVER for permanent ink but as this is a US site might not be available here. You could also try whiteboard ink remover - some are aggressive as hell.
It works on the range balls I've picked up on the practice ground where the pro has written range in blue marker or sharpie pen (normally when I find Pro V's he has just transferred from a competition ball in his bag to use as a practice ball)