Would epoxy be strong enough to bond a piece of slate to the bottom of a vertically standing piece of driftwood?
I was also thinking drilling it to the bottom instead.
If the piece of wood is small, I would use epoxy, but if it's a bit larger, I'd definitely use a stainless steel screw just to screw it in (or even several screws)
I've had luck using some aquarium silicone to attach smaller (10" x 6") pieces of driftwood to slate bases. Most of the bases I see commercially use a single stainless steel screw; I just happened to have a bunch of extra silicone lying around.