There is often a cold water tank in the loft - two tanks - one for domestic hot water and another for the central heating radiators. Your overflow may be coming from one of these or from a WC cistern.
Whichever it is - the cause will likely be the need to adjust the inlet valve to shut the water off sooner - or to replace the perished rubber seal inside the valve. The first is easy and so is the second but the mains water into the problem valve needs to be isolated first.
You will do well to try and trace / determine the source of the overflow pipe i.e. where does the dripping pipe go to?
Good luck...
Peccavi
May 2010