What geological processes would 'restart' subduction processes? According to the Science Daily article Plate Tectonics May Grind To A Halt, Then Start Again (Carnegie Institution), one main aspect of the research being presented is the hypothesis that


  suggests that plate tectonics may have ground to a halt at least once in our planet's history -- and may do so again


Specifically, the article states that geochemical evidence of igneous rocks from about a billion years ago suggests there was a lull in subduction processes, coinciding with evidence of the closing of a Pacific-like basin, forming the supercontinent Rodinia.  The lulls also, according to the article's authors, in part explain that


  By periodically clamping the lid on heat flow, intermittent plate tectonics may explain why the Earth has lost heat slower than current models predict. And the buildup of heat beneath stagnant plates may explain the occurrence of certain igneous rocks in the middle of continents away from their normal locations in subduction zones.


However, the article rather vaguely states that


  Rodinia eventually split apart when subduction and plate tectonics resumed.


What geological mechanism would cause a 'restart' of plate tectonics, specifically subduction?
Subduction is ultimately driven by the heat flow process of mantle convection. A mantle plume beneath a stable plate will ultimately cause that plate to split, a la the African rift zone we observe today. 

The crust of the earth is very thin compared to the mass of flowing material beneath it. As my global geophysics prof taught us, the fact that the earth has a solid outer "skin" is a consequence of the thermal conditions around the earth's surface and geochemistry, but that has little to no effect on the thermal conditions within the planet that ultimately control thermal convection that expresses itself on the surface as plate motions. I.E., the thermal mass of the crust is inconsequential when considering the thermal mechanisms that drive the mantle and outer core. Heat makes the plate go! 

On the surface, plates may move, be created or destroyed, but thermodynamics tells us these are consequences, not drivers. We may observe evidence of past situations where plates appear to be static with respect to the underlying mantle motions, but this does not reveal any new mechanism that would suggest plate tectonic theory would not predict such circumstances.