When drilling beach stones, I would suggest you avoid the advice about buying the best drill bit you can find. When drilling stone, diamond bits wear out quickly, whether you buy the twisted ones (expensive) or the straight ones (far less expensive.) So go for the straight ones.
I'd suggest you go for the economy diamond cylinder drill packs from Rio. They come 25 in a pack and cost less than a buck each, compared to $2 or $4 each for twist bits. You can get through five or six stones with a single bit if the stones are not too hard.
Start your hole with a diamond ball burr, say 1 mm Hold the stone underwater with your hands in a small plastic bin, like tupperware or glad disposable.Make a small, round divot in the stone.
Once you have a achieved a small divot, drill through the stone in with your cylinder diamond drill bit. Unless your stone is very small, avoid any bit under 1 mm. The .75 mm bits have a tiny shaft and break often. The 1 mm bits have a reasonably strong shaft.
To make sure you don't drill through the bottom of your container, put a couple of popsickle sticks under your stone. Hold the stone under water, on top of the pop sticks. Use clear water so you can see what you are doing. The top of the stone should be just barely under water. As you drill, a long tail of dust will drift off from your stone in the water. Press a little, pull back a little , get a rhythm going. When you get 4/5ths of the way through, turn the stone over to see where the hole is going to come out. Mark it with a sharpie, knick that place with a ball bur and start drilling from the back. When the hole meets the other side, everything else becomes easy.
Widen the hole with successively larger (cheap) diamond drill bits, drilling under water. I find a 2.5 mm hole holds a nice rivet and makes a nice pendant.
cheers,
Shawn
(who has made 8 jeckyll island rock pendants today for our company's united way auction and has cramped hands.)