AI am doing a cake with a washing line on the side and want something edible which we can tie at the end with some hanging down. Any ideas? White liquorice string would be ideal but I don't think it exists,

buttercream piping is an idea--

or extrude some fondant through an extruder or pasta machine

AIf you have access to some, you could pipe some sugarveil/cake lace? X

why not string?  You can eat string.  It just meanders through until it exits. Like roughage.

Baby candy lollypops have string handles on them.

AI'd just use a thin cord of gumpaste.

Extruded fondant would work if you can fake the knot.  You should be able to let some hang freely and it'll firm up.

If you need it to maintain its flexibility, you might be able to use homemade gummy (I have a recipe here: http://www.eat-the-evidence.com/gummy/basic-gummy-recipe-and-tutorial/ ) but you'll need a way to mold a nice round string.  If you don't have a mold, you could use some ComposiMold (the food safe kind) and make a temporary mold of some string, then put the gummy in that.  If you need the line to actually support weight, I'd suggest adding an addition packet of gelatin to the gummy mix for higher tensile strength.

This is actually a really cool problem and is setting my mental wheels of experimentation in motion... :D