Is the computer dead during the glitch? Can you see it over the network? If not, you might need a new motherboard, since I'd suspect the graphics chip.
If the Mini is still running normally, just with a glitchy screen, I think it's probably a broken connector/cable between the motherboard and screen, but the screen itself could be broken. I don't think there's much you can do without spare parts to try. You could try disconnecting and reconnecting the cables, but they're very delicate and it's easy to make the problem worse.