This problem can be resolved in several ways...
1. New board (easiest, but most expensive)
2. New power button (fairly straight forward, de-solder and solder new switch, costs a couple of quid)
3. Clean existing switch (quite complicated, costs nothing)
I think there is some information I gave out on here a while back. I've fixed a couple of the minis belonging to guys at NPF with the same issue...
I'll have a bit of a search and when I find the info I'll PM it to you and post it in this thread so it's available to the masses.
If you want me to fix it for you, it's not a problem. Just get in touch, as long as you pay the postage each way, we'll work something out for the fix.