In electrolysis, why do we keep the electrodes the same?

If the electrolyte contains chemical species that will be reduced at the cathode or oxidized at the anode (or both), that means chemical change.  The electrolyte can form one or more compounds or ionic species that weren't there before.   
 A given species could form at one electrode but diffuse back over to the other electrode where it promptly breaks back up into its reactants. 

If the electrolyte does not participate in any reaction(s), it will just act as a conductor of electricity.  The same goes for the electrodes.  Some materials will react but will not produce anything obvious (gases, precipitates, or color changes).  Others will not react at all under the specific combinations of voltage and current.