Check the cable's free movement by tugging on the exposed inner wire alongside of the bike's top tube or down tube. You'll be looking for perfectly smooth derailer actuation and return action.

Old lubricant eventually becomes thickened.
Often some thin-bodied lubricant like Tri-Flow introduced slowly into the end of the cable housings will flow in and free up a sluggish cable movement.

And I'll second bwilli88's recommendation to correct the cable tensioning, such that the shifter can move the derailer all the way to each extreme of sprocket positions.
Often the shifter's clicks don't position the derailer centered accurately with each sprocket position, resulting in noise, which can be fine-tuned using the cable's barrel adjuster.

But also first verify that this "derailer" problem is not a case of a shifter that does not actuate (click) or won't hold gear position reliably.
And inspect each end of all cable housings, to check for having possibly pulled out of a ferrule (end cap) and then having not fully seated back in.