How to show the direction of highlighted path in graph

You can add additional graphics (arrow) using Epilog or Prolog:

Ex1:

HighlightGraph[g, path, 
 Epilog -> {Red, Thickness[.01], Arrowheads[.05], 
   Arrow /@ Partition[GraphEmbedding[g][[path]], 2, 1]}]

Ex2:

HighlightGraph[g, PathGraph[path], 
 Prolog -> {Orange, Thickness[.005], Arrowheads[Table[.04, 10]], 
   Arrow[BSplineCurve[GraphEmbedding[g][[path]], SplineDegree -> 2]]}]

This is a V9 answer:

g = DirectedGraph[GraphData["DodecahedralGraph"], 
                  VertexLabels -> "Name", EdgeStyle -> White];
HighlightGraph[g, FindHamiltonianCycle[g][[1]] // Most]

g = DirectedGraph[GraphData["DodecahedralGraph"], VertexLabels -> "Name"];
HighlightGraph[g, FindHamiltonianCycle[g][[1]] // Most]