Find yourself an engineering mechanical pencil that has a very small eraser and clean the contacts on the remote that come into the contact with the docking station pins. They may be corroded. You can also power off the remote and try using a cotton swab (q-tip) with either 70 or 90% isopropyl alcohol.
From what you describe it is either a corrosion issue on the contact board or a bent/recessed pin in the docking station contacts.
Last thing i would suspect here is the need for a new battery but might not hurt to try another one as the regulator circuitry that is inside the battery plastic wrap might also be going bad.