There are not free electrons in water, it is the ions in the water that carry the current... From the Wiki...
Electrolysis is the passage of a direct (http://en.wikipedia.org/wiki/Direct_current) electric current (http://en.wikipedia.org/wiki/Electric_current) through an ionic (http://en.wikipedia.org/wiki/Ion) substance that is either molten or dissolved in a suitable solvent, resulting in chemical reactions at the electrodes and separation of materials.
The main components required to achieve electrolysis are :
An electrolyte (http://en.wikipedia.org/wiki/Electrolyte) : a substance (http://en.wikipedia.org/wiki/Chemical_substance) containing free ions (http://en.wikipedia.org/wiki/Ion) which are the carriers of electric current (http://en.wikipedia.org/wiki/Electric_current) in the electrolyte (http://en.wikipedia.org/wiki/Electrolyte). If the ions (http://en.wikipedia.org/wiki/Ion) are not mobile, as in a solid salt (http://en.wikipedia.org/wiki/Salt) then electrolysis cannot occur.A direct current (http://en.wikipedia.org/wiki/Direct_current) (DC) supply : provides the energy (http://en.wikipedia.org/wiki/Energy) necessary to create or discharge the ions (http://en.wikipedia.org/wiki/Ion) in the electrolyte (http://en.wikipedia.org/wiki/Electrolyte). Electric current is carried by electrons (http://en.wikipedia.org/wiki/Electron) in the external circuit.Two electrodes (http://en.wikipedia.org/wiki/Electrode) : an electrical conductor (http://en.wikipedia.org/wiki/Electrical_conductor) which provides the physical interface between the electrical circuit (http://en.wikipedia.org/wiki/Electrical_circuit) providing the energy (http://en.wikipedia.org/wiki/Energy) and the electrolyte (http://en.wikipedia.org/wiki/Electrolyte) Electrodes of metal (http://en.wikipedia.org/wiki/Metal), graphite (http://en.wikipedia.org/wiki/Graphite) and semiconductor (http://en.wikipedia.org/wiki/Semiconductor) material are widely used. Choice of suitable electrode depends on chemical reactivity between the electrode and electrolyte and the cost of manufacture.