Silicone grease of rather thick consistency (NLGI class 2), provides both lubrication (preventing fast weardown) and mechanical dampening (lowering mechanical snapback enough that the remainder is easy to eliminate with the PhobGCC stick filters).
Comes in a syringe with an extra short 0.84 mm inner diameter / 1.27 mm outer diameter tip that makes application very easy, as well as a small silicone cap to prevent leakage. Each syringe contains 5.0 ml of silicone grease, which is enough for around 200 stickboxes.
Made in Germany, product link: Liqui-Moly Silicone Grease
While this grease can also be used with stickboxes on OEM controllers, it needs to be applied very sparingly, as applying too much of it will make it creep into the stick potentiometers over time, messing up stick inputs.
On PhobGCC, no stick potentiometers are used, and should the grease creep up to the magnet holders or magnets, that doesn't cause any problems – assuming that you've properly glued the magnets to the magnet holders as well as the magnet holders to the stickbox rails, as instructed in the PhobGCC build guide.
Especially when using older controllers (2008 JP white or earlier) as the base for your PhobGCC build, I highly recommend lubricating your stickbox with this stick grease after deep-cleaning it (full disassembly and washing in a sink with warm water and dish soap).