Join Date: Feb 2019
Location: Los Alamos, New Mexico
You want a non-calcareous stone. That obviously rules out limestone, marble, or coral, but it also rules out some forms of sandstone and metamorphic rock. You also want to avoid anything likely to have heavy metals. This pretty much means a silicic stone, probably volcanic or metamorphic.
This is where I'm not sure how much I can help. Stone sellers have their own terminology for stone varieties that doesn't really map to scientific geology.
However, "lava rock" will usually mean a silicic volcanic rock that should be okay, so long as it isn't calicified (has not had lime deposited in it by groundwater.) If the "lava rock" doesn't have any kind of white rind or white infilling in its pores, it's probably a safe choice.
Slate is usually safe, too, which is why it is popular. I understand that aesthetics matter.
Anything described as a "granite" is probably safe as well.
If you have any particular varieties you've come across, post here and I'll see if I can figure out what it actually is.