Thicker headgaskets lower the compression, and make it safer to run higher boost, but lower the responsiveness of the engine when off boost
I doubt the change to lag would be very noticable at all.
Depends on what kind of turbo you're looking to run really, if you're talking 1.4-1.6+bar on a TD04 or something, then you'll want a thicker headgasket, but if you're only talking 1-1.4 bar on a Hybrid or similar, you'll not really
need the thicker headgasket. For example, Toby on ToyTuning uses a TRD headgasket, which is around 0.6mm on his hybrid turbo built engines to retain off boost performance and to get a higher power figure when on boost than you would with, say, a 1.2-1.6mm headgasket (due to the higher compression achieved)
Dave