Replaced the batteries with fully charged rechargable batteries. The Might Mouse and now it is fully functional.
Rechargable batteries can be a problem with a Mighty Mouse.
Although the battery level was at 40%, the Mighty Mouse started to loose some of its functionality. If you choose to use rechargable batteries, as I encourage you to do, replace them before the battery level drops below 50%.
It is hard to see why using rechargeable batteries should pose a problem. And replacing them before the charge level drops below 50%? Ludicrous.
Anyway, I have the same problem with the mighty mouse (Apple's first servicable mouse), using normal batteries, after upgrading to Lion. No problems with either left or right clicks under Snow Leopard. Most annoying.