You may receive some lodestones (expect only a few per thousand items), when you recycle items. The rarity of an item doesn't seem to play a role (i.e. recycling one common item gives the same chances to get a lodestone as recycling one legendary or exclusive item, but gives fewer recycle points).
Lodestones are raw resources (not gems) required to build Wonders . You find out how many you have, go to the raw resources tab of your inventory (you need to have at least one for "lodestones" to be listed.)
Lodestones are not shared between islands, if you have multiple islands.
Lodestones can be found in crates and barrels too.