Have a wire of this colour sitting loose by the starter BUT also an odd one in the wire loom in the engine bay (one near the horn relay) that is (faded) perhaps white with slate stripe and blue twirl (fancy..) which UK wire chart seems to say is to do with tachometer. but one in loom sits idle not connected any place.
also notice that the hexagonal bolt thermometer (??) on side of radiator has two short cables coming from it and i have attached heads to them but there already seems to be a water temperature wire in this loom going to the water pipe on the engine block top beneath edge.
so can't figure if these wires are simply redundant (maybe I recall they were a trigger for an additional radiator fan to come on at certain temp??)

 

There is a wire in the loom which is white with a blue stripe that is located next to the start relay on the bulkhead which is unused. When I first got my Series 3, the electrics were in a bit of a mess and I didn't know where this wire should connect to, but then when I got the official wiring diagram it is shown as " not used". See P9 from the wiring diagram below as a pdf file. I have not checked to see if the other end of this wire comes out by the starter.

The standard temp switch for the electric fan relay is on the RH side of the radiator. Perhaps the one on your car has previously failed and someone has fitted a different temperature switch elsewhere?

 

I have no experience with Series II stuff but the "not used" white/blue shown in Andy's Series III diagram is for the 3.4 cars with points ignition. It bypasses the ballast resister when the key is turned to "start". I suspect a Series II could have the same or similar arrangement

Cheers
DD