The solenoid is a coil of wire that sits on top of the oil pump - when it is energised it opens a valve inside the pump, allowing the oil to flow. As elenar says, these solenoids are prone to failure, and cost around £22 - quite expensive as a pump complete with coil costs around £50 (Danfoss). I think your boiler needs a thorough clean - excess soot in the flue can cause all sorts of problems and at least no parts are needed! I'd also change your boiler engineer.....!
John (OFTEC), September 2009