When battery operated clocks stop working, it's usually caused by one or more of the batteries. Either a battery has lost its charge, or battery acid has leaked, causing corrosion. Often, the internal mechanisms, such as the cogs, springs and other movements of the clock, are not faulty and are unaffected. Repair of a battery operated clock typically comes down to the batteries and the battery compartment, making the fix simple.
Open the battery compartment with your fingertips and nails or use a screwdriver to open it. Inspect the battery compartment for corrosion -- you will likely see a "bloom" of blue and white or green and white mold around the ends of a battery.
Put on gloves and pull the batteries out with tweezers or tongs. Alternatively, pry the batteries out by inserting the blade of a screwdriver underneath them and pushing down on the handle.
Clean the battery compartment thoroughly with acetone and a wire brush. Get the battery leads completely free of corrosion; wipe dry with a cloth. Let the compartment air-dry for about 10 minutes; any residual acetone will evaporate.
Put new batteries in the compartment and set the clock to the current time. Close or replace the battery compartment door and put the clock back in its place.