Yes, a potato(e) can be used to power a lamp. Usually a penny & a galvanized nail are used for this 'battery'. Copper & zinc are REQUIRED for this process, but not a penny & a nail per se. A potato works well, but a tomato, lemon or other citrus fruit can be substituted. The zinc and the copper are the anode and cathode terminals of your potato battery. Using ordinary hook-up electrical wire, you can use the potato to create a voltaic cell, which will power a VERY small bulb. A light emitting diode (LED) will work fine.
Electric current is the movement of electrons from one atom to another in a conductor. Inserting the two common metal electrodes into the potato causes a chemical reaction to occur resulting in current. The potato does not participate directly in the reaction. It is there rather as an electrolyte to facilitate the transport of the zinc and copper ions in the solution, while keeping the copper and zinc electrodes apart. The potato contains phosphoric acid (H3PO4), which facilitates the electro-chemical reaction of zinc with copper.