The bone in edit mode has no rotation. And that probably why you cannot obtain a local axis coordinates.
But you can set the axis to "normal" and (for instance) use 'individual origins' as pivot point to rotate it around X or Z.
To rotate it around Y, use the roll value on the panel on the right.
You can do it with 'local' (or normal) axis in pose mode. It will work for X, Y and Z (no roll here). Once done, apply the pose as 'rest pose':