I wouldn't bother; one of the nice things about silver, electrically speaking, is that its oxide is quite electrically conductive. I'm a jeweler, and I don't have anything that would easily clean the hole on a binding post. Anything you "might have around the house" is probably going to be unnecessarily abrasive (emery board, sandpaper, et cetera), or leave a residue behind that's even harder to remove than the tarnish (any kind of metal polish, kitchen cleanser, or whatever with a pipe-cleaner, q-tip, or toothpick, for example)... or both!
It's silver; it reacts to sulfur in the air, and oxidizes. C'est la vie.
If it really bothers you that much, get a "Sunshine Cloth" - they're a buck or two on eBay, and are the absolute best polishing cloth for precious metals out there, bar none - and figure out some way to get it, or a piece of it, into the hole on your binding post. Or replace the post with one that's gold- or rhodium-plated, and thus won't discolor.