I'd check/clean the nut, the bridge and the stop tail-piece, especially the tailpiece posts. Loose posts can create serious tuning havoc, as can sticky nuts and bridges.
To check the tuner, simply swap the B and G string tuners and see if the problem persists. If the problem continues with the G string, it isn't the tuner, if the B string starts going out of tune often, it is the tuner.
Also, of all the strings, the G string tends to lose tuning first in the presence of aggravating circumstances - such as too high action. Many low-tension, ultra-slinky strings are simply not suited for high-action guitars, and assume use on extremely low action setups.
You may want to try switching to a thicker string gauge - the Les Paul itself was designed for a tone that usually assumes a thicker gauge than you'd use on a strat. While I use 008s and 009s for strat-style guitars, I prefer 010s or even 011s on my Ibanez AR50 (that has a very LP-ish tone/feel).
Of course, all this assumes that you are using good strings to begin with. I didn't want to start with this point - I assume anyone using an LP wouldn't cheap-out on strings.