Here is a basic video guide to promoting your elemental relic. Each time you get the growth level to a filled 9+one more relic (which makes it's level technically 10 even tho it still says 9), you can promote your relic to gain a relic level. The relic shown here is way over due to be promoted. In fact, I think I lost quite a bit of my applied exp due to the promotion, so don't wait as long as I did! xD
When I put it into the relic menu for promotion it gives me two possibilities. I am unable to influence this level's possibilities. But, as you can see later, I will be able to influence the next levels - just 10 growth levels to go before I can do that :S
Although I got my influencing item for free from a quest, you can buy them from the shop behind the Rose Queen. It requires trade gold or stones of awakening for these items. Stones of awakening can be purchased from second Otis in Nightfall with tokens of faith.