Bad luck. The MacMini has a highly integrated logicboard with the video chip soldered onto. You'll have to replace the entire board.

My mac mini is also acting up (the display randomly stops outputting and you have to wiggle the connector) and like the poster before me said, it is pretty much impossible.

I've had the board out and done a continuity test from each pin of the video output back to the board but after that the whole thing is lacquered and multi-layered, making further tests impossible. I also checked the whole board for small cracks and other signs of physical damage but on the whole I didn't find a thing.

I don't think theres too much hope for problems of this kind.