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Does Quantum Mechanics affect this at all...

My brain hurts!

But seriously I was trying to answer more generically. I think you're right. A bound electron would end up in a finite energy state - it's minimum or zero point energy state. Off the top of my head, I don't remember exactly how to treat an unbound electron, but the uncertainty would put a non-zero minimum limit on its energy.

I'll telepathically summon the master.

[AndrewC]

Does Quantum Mechanics affect this at all, since an electron's momentum (a measure of speed) and position are decribed by probablity functions?

Which only become "real" when "measured".

[Physicist]

How does the third law of thermodynamics prevent an electron from being at rest? Be careful when you apply statistical arguments to single objects.

That, of course, segues into the real reason an electron cannot be at rest: the Heisenberg Uncertainty Principle. If an electron were at rest, it would be at a well-defined location with a well-defined momentum.