To test the possible friction along cable length, I would disconnect the cable from the derailleur side and pull the cable, either with the shifter or by hand on the derailleur side. Besides the friction along the cable, the culprit could be hardened rubber of the shifter's grip. Then, without gripping with a large force, the hand just slides over the grip, creating general impression of the shifter being ineffective. While replacement grips may be available commercially, their cost may be higher than the value left in the shifter.
For the interior of the shifter, silicone grease may be used. I doubt, though, it could help much. If the interior were causing excessive friction, I would expect the cable to be ran incorrectly through that interior.