Does the bending of a tree's trunk in the wind stimulate and strengthen root growth?
In tree growth there is principle called The axiom of universal stress whereby the growth is in such a way as to equalise the stress across the whole of its structure. The roots, the stem and the branches are one continuous system and can not be thought of as separate.

On a simplistic level for a branch to grow, there must be a corresponding equal and opposite growth in the stem and roots. Of course it is not quite like this as the growth happens across the whole structure at the same time.

The trees on the edge of an even-age stand of trees will have grown to be more stable against wind. They are referred to as being a wind-firm edge to the stand. The trees that have grown up within a stand, with the shelter of their neighbours will not have the same stability in the face of wind, so that if the stand is opened up too quickly it can lead to wind-blow. In uneven-age woodlands, the same principle is true but the dynamics make it less pronounced.

The reason for mentioning this is just to illustrate that, absolutely the wind, and other external factors, gravity being the obvious one, will have an effect on the growth of the roots, as well as the rest of the tree. As the wind exerts a force on a tree that  force will travel through the structure and into the roots and into the ground.

To come back to the specific question then.


  Does some kind of mechanism exist within the tree that connects mechanical bending of the trunk with increased rate of root growth?


Yes, and no. Because the increased rate of growth will be across the whole structure, but not equally. It will put on growth at the points where the stresses are, in order to self-optimise the loading including within the root system.

The next time you cut either the stem of a leaning tree, or a branch of a tree, take a look at the orientation, and thickness of the rings. You will see that in broadleaf trees (or more correctly angiosperms) the rings on the side under tension will be thicker. In conifers (or gymnosperms) it is the opposite. The thicker growth is found on the compression side.

Here is a great photo of compression wood in a conifer from http://www.copperman.co.uk/didgeridoo/how_to_make_a_wooden_didgeridoo/what_is_wood_2.php



The same principle will be happening in the roots also.

As to the actual physiological mechanism that causes this response I have not yet come across a definitive answer. The consenus, at least as far as I know, is that it is not fully understood. 

The closest I have come across so far is from this phd in 1994. http://etheses.whiterose.ac.uk/2438/1/DX184141.pdf

I dare say there has been more work done on this in the last couple of decades.