After a little practice it gets easy to quickly identify the mid point of any LOS from hex center to hex center, this is usually the middle of a hexside. Hexsides, I dunno.
Almost: the mid point of a LOS from hex center to hex center is always either a hex center (easy to figure out) or the middle of a hexside. See sketch of proof below
I suppose the hexsides would create a equilateral triangle grid where some smart person could calculate the exact midpoint, but hey, let’s keep it real until either I figure a easy way or somebody else explains it to me. Dead reckoning works pretty good, less thinking.
That's it exactly. Each hex could be divided into 6 equilateral triangles, so hex centers and vertices are actually the vertices of a triangular grid. So you could give each vertex and hex center a pair of coordinates (preferably in some non-orthogonal system, unless you like computing with square roots instead of integer numbers) and compute middle points this way. In the right system, each pair of integer coordinates will correspond to either a hex center or a vertex; but figuring out exactly which are hex centers and which are vertices takes a bit more work.
Practically, here is how I do it: start (mentally) with a finger on each end of the LOS, and move both fingers incrementally, in symmetric ways in the general direction of the midpoint; this keeps the midpoint of your two fingers fixed. Once they're close enough, you can figure out the exact midpoint. (If you do this while playing FtF, most opponents will be puzzled; on VASL, it's safer)
BTW, this is one way you can prove that a midpoint is either a hex center or a hexide middle: this process will end either with two fingers at the same hex center (in which case this is the midpoint), or with them in adjacent hexes (in which case the midpoint is the middle of their separating hexside).
The midpoint of a hex center to vertex LOS is a more complex beast, and could be another vertex, or the middle between a hex center and its vertices.