For anyone who has completed the White Mage Quests, Thyrus needs no introduction. It was the staff of A-Towa-Cant, your predecessor in White Magic and the same padjal whose garb you receive. The staff is made of petrified wood rather than a more mundane material, which gives it the appearance of stone. Speaking as a Geologist, that's some serious nature-power going on there.

The Artemis Bow belonged to Gilbert the Godsbow, most famed virtuoso in Gridanian history. It is said he had bow crafted to also be an instrument. The tuning keys on the harp also serve to fine-tune the bow, as changes in humidity can cause the wood to swell and warp over time. It was preciously kept at the God's Quiver until it got ransacked by a bunch of Ixal and the rest is history.
Your own mentor, Jehantel, appears to have an Artemis Bow as well, but it is not explained how. My guess is that it is a replica he had crafted himself to his own standards (or that of the moogles), although it is unlikely as practical as your relic reborn.

The Omnilex is a relic of the Nymian scholars, said to contain all the wartime battle tactics they had ever conceived. Many copies were made, but few have survived the test of time. The metal decorations on the bindings were also designed to help the scholar channel their aether when casting, making the tome far more useful than a simple treatise on battle tactics.

I can also understand why we aren't given Oathkeeper as our relic. The story would see us obliged to return it to the Sultansworn and failing to do so would not be very Paladin-esque. It would also wrap up the gaps in the Paladin quests, which would be a premature thing to do from a design and development standpoint.