This question is in reference to the use of the word "hand" in "right-hand side" (and applies equally to the left).
My question is what does "right-hand side" say/imply that "right side" doesn't?
Handedness is perhaps the most common way of teaching someone left from right. While there are other natural things that exhibit consistant directionality (the circular flow of water in a drain in a given hemisphere; the rotation of the earth when seen from above a particular pole, etc.) none is as readily accessible to the viewer as her or his own handedness. Most of us know from an early age which of our hands is dominant, and that is labeled by our elders as either right or left, depending on our proclivities (let's skip the brain dominance discussion for now).
We can then identify whether the direction or side we are considering corresponds to our right or left side, based on our knowledge of our dominant hand.
I think the hand reference when discussing side is a reflection (residual) of this basic analysis we learned in our youth.