The answer will depend on how one defines the distance from a point to each of the sides.  For some sides, the perpendicular from the point to that side intersects the side.  But for others, the perpendicular will strike the line that corresponds to the extension of the side; outside of the hexagon.

So for those point and side combinations, one might mean the distance to the nearest vertex of that side.  

For example,

the vertices are:

(1,0),(.5,√3/2),(-.5,√3/2),(-1,0),(-.5,-√3/2),(.5,-√3/2)

How far is the point, say, at (0.9,0) from the segment (.5,√3/2),(-.5,√3/2)?  Is it the perpendicular distance to the line y = √3/2, i.e. √3/2 or is it the distance from (0.9,0) to the vertex at (.5,√3/2)?

If we are always measuring perpendicular distance to a side or to a line extension of the side, the average distance is always √3 / 2.  Imagine any point inside the hexagon and pick one pair of parallel sides, say one is the line y = √3 / 2 and the other is y = -√3 / 2.  Those 2 distances will always sum to √3.  Now rotate your head 60 degrees as you look at the paper.  Same situation.  Summing all 6 sides gives 3*√3