What [if anything] is special about the boiling process of purifying water (excluding temperature)? I know that water boils at $100~^{\circ}\rm C$. I also know that the standard way to purify water is to boil it for roughly a minute. What I'm curious about is, what properties of the boiling process purify the water? 

Is it purely a temperature thing? Are we saying "roughly a minute" so that the temperature actually increases beyond $100~^{\circ}\rm C$, thus reaching some actual temperature needed to kill bacterial/viral stuff? Or, does the "rolling" of the boil actually add a physical benefit to the purification process?
There nothing special about boiling that is doing anything.  For example, you can put a beaker of water under vacuum and get the water to boil at room temperature, but that won't kill any bacteria or render any viruses impotent.

It's purely a heat thing.  You want to destroy any pathogens that may cause illness.  Sufficient heat will denature proteins in a virus and kill bacteria, etc.  Boiling simply gives you a visual cue that you've reached a very high temperature (so you don't need a thermometer) and the ~1 minute is just to ensure that you sustain a high temperature long enough to destroy any bacteria or viruses.