You can probably get a single masonry bit for not too much. Or if you have a dremel, you could try using one of the grinding or diamond bits, but that would take a lot longer. Try to keep the spot you are drilling wet, or even drill with the piece of slate submerged. It keeps the dust down, and makes it a bit easier to drill through.
I'm fairly certain I once drilled through a piece of slate with just a regular high-speed bit (granted, it pretty much ruined the bit), so if you have a regular bit you don't like...
Anyways, I really like bolting driftwood onto slate, aside from just holding the piece down, it also allows you to place it in positions that gravity alone wouldn't support. I don't think there is too much to be concerned about with the metal in the screw, but I'd at least avoid brass. I use nylon machine screws, but I happened to have a 1/4" tap lying around, so I drill and tap a hole in the driftwood, and then just screw into the wood. If it's a piece you can drill through and keep covered in substrate, you could but a nylon bolt on it as well.
Probably just easiest to use a wood screw though.