I don't know the historical origin of the diode symbol, but I imagine that the arrow-like symbol is intended to represent the one-way flow of current through the diode (in its simplest ``one-way-valve'' model). The arrow points in the direction of forward bias (positive to negative). With bias in this direction, the current flows in the arrow direction.
(Note that for a Zener diode in avalanche mode, the current flows in a direction opposite to that of the arrow!)