STI shifter sometimes doesn't catch in cooler weather
WD40 will work, for awhile, but in the longer term it'll work against you since it's a poor lubricant and will work to prevent future lubing. Spraying some penetrating oil into the ratcheting mechanism is the way to go, long and short term. Gaining access by working the brake on the shifter and spraying, using a long neck attachment for better aim, into the little hole, opposite the cable side, that you can see when applying the brake, leading to the gear ratchet will do the job. Zimm's confirms this, as does my local bicycle shop.

I just did this on an Ultegra gear shifter, using "Super Lube", and now the shifter works like new. After it evaporates it leaves a extremely thin layer that is highly slick.