Crystal
Olivine from the mantle, from a Red Sea island, and occasionally from a meteorite.
These are associations the community holds, not settled fact — the more a reading is shared, the bolder it sits.
Peridot is gem-quality olivine, and it is one of very few gemstones that occurs in only a single colour — the green is intrinsic to the iron in its structure, not a trace impurity, so there is no such thing as a red or blue peridot. Olivine is a principal mineral of the Earth's upper mantle, which means peridot crystals have generally come up from depths that most rocks never see, carried in basalt.
The historic source is Zabargad, a small volcanic island in the Egyptian Red Sea, worked for perhaps three and a half thousand years and at times kept deliberately obscure — Pliny reports it was patrolled and that the stones were guarded. Its name in the classical world was topazos, which is why some of the "topaz" in old texts and old church treasuries is in fact peridot; the great green stones in the shrine of the Three Kings at Cologne Cathedral were long called emeralds and are peridot.
It also falls from the sky. Pallasite meteorites are iron-nickel matrices studded with olivine crystals, and some of those crystals are cuttable. Practice reads peridot as lightness and unburdening, which fits a stone this bright, though the reading is recent and the older lore is chiefly about the difficulty of getting hold of it.
Readings practitioners have offered, ordered by how many recognise them. Take what serves, leave the rest.
Carried forward
Pliny the Elder, Natural History, Book 37 (c. 77 CE)
Pliny places topazos on an island in the Red Sea, found by pirates who were half-starved and digging for roots, and says the island was so often lost in fog that it took its name from the difficulty of finding it. He also notes it is soft enough to be worn away by the file, which no true topaz is.
Carried forward
Common in contemporary practice
Associated with setting something down — resentment most often. This is a twentieth-century reading with no classical warrant, and it sits alongside a far older and better-attested reputation as a stone that was simply very hard to obtain.
Credit what you carry. Much of this comes from living traditions, and some of it from traditions that are closed — held by particular peoples and not offered for general use. Where you know whose practice a reading came from, name them; where you don't, say that too.