Ok, you know the main CoO room, where you start the raid and where the hall of statues are, there are two adjacent rooms, "down hill". You blow up shield cores in there during early TTK missions, and some of the last relics for the first part of the raid spawn there. During patrols, those two adjacent rooms can randomly spawn relics. You just grab the relic (a one minute timer starts counting down as soon as you pick it up), run it back to the CoO statue hall, and you can deposit the relic to a statue. I deposited it in the statue the furtherest away from CoO because it was the first one I approached and it spawned a Tier 3 CoO session for free. It didn't cost a rune, the relic acts as a rune. I didn't check the other statues while I held the relic, but my guess is you might be able to deposit it to any tier statue you like. This seems to be a method to make sure even if you run out of runes, you have an option to still run CoO to get more runes.
Edit: You know what, I need to check this again. My inventory was a mess, and I might have not had my antiquated rune count accurate. I am pretty sure it said, "deposit relic" but it could have just taken my rune even though I was holding the relic. Er... possible theory.. needs more testing!