That's a really tough question.

side in what sense? side of a body? side of a river, etc? side in a battle?

I'd say má without complication in most cases. However, the 'side' as part of body I cannot even predict.

+James Coish as in a lateral surface of an object, body etc. A flank.

Something tells me the root PAL could be of use. palme "surface" is not very far (cf. E "side" G "Seite" coming from a root meaning "broad").
Or the root AR of course, though I don't see yet how

I first thought flank as I was looking up the etymology of side.

*ranco+nemno/na, +nemba (< NEB) for the sense "rib"?

In the sense you're going for, I would agree with PAL.

Does *ar(a)palme "beside-surface, outer surface" sound very contrived?

I don't think derivatives of √PAL could be used for surface that surrounds three-dimensional objects (all around); IMHO they only refer to two-dimensional, horizontal relations and entities.
Also "ad-/ar-" only puts you to the space close to the surface, but in any side of it (front, back, above, below...), not specifically in the flank sides.

+Lokyt L. you may have a point, although as the language develops, the original stricter association of PAL with horizontal surfaces may become blurred and meaning could become more generic