It depends on what type of outer the cable has, if it is a Nylon lined then it is unlikely that lubrication is a long term option.
Older aircraft have cables that are a metal coil outer with the cable running through the centre, these can benefit from lubrication, Alioth suggests LPS-2 and I would not disagree however I have used Mouse Milk to free these cables if they are very difficult to move.
I don’t believe anyone pretends there is a permanent solution. But as with so much in aviation maintenance, if you can safely get another 10 years out of something, why not? And, as I said, the labour cost can be really substantial getting access to some cables.
I maintain my plane on a money-no-object basis (it is the cheapest way to run a plane) but will still grease a control cable, especially a non-critical one for the heater 
I am changing my prop governor cable at the next annual. It has been running against some part of the engine, and the outer sheath has come off in places.
I happened to be talking to a chap with hire boats at a marina. And he said that since starting yearly lubrication of the control box / cables they’ve not had any failures.
Although they pulled the unit off, and used what looked to be a length of hose which they filled with oil then put in a small airline at low pressure to flush them right through each cable. It was not a quick process.
The TB20 propeller pitch control cable is €442 + VAT
So the original Q is very valid. There may be nothing wrong with it. However, we can probably extract it at the Annual and get inside it properly.
Most likely it got tight because rain water gets in at the front – as the above pic shows. It’s a crap design; there should be some sort of rubber shroud. However, this seems common in GA; the other day I saw a plane with a totally exposed alternator right at the front. The alternator must just fill up with water every time it rains.
A set (throttle, prop, mixture) of PMA cables for a M20 is said to be around $1000.
