[@Vec] For the secrets things, I was thinking around the lines that when people wished to hide things or make them disappear, whether physical evidence or knowledge they wanted to remain secret, they could give it to Khthon and he would keep it hidden. So like, his thing would be hidden wealth and storing secrets, and to a lesser extant hidden truths and forgotten knowledge. He can't control who discovers secret knowledge in general, only the secrets he specifically protects. I also really like the idea of him being able to hear all secrets whispered underground. Before Mortals come around, he'd probably use the Secrets Domain to hide himself and his treasures from the other Gods when he doesn't feel like dealing with them.

For offerings, most things would work if done sincerely. Like mentioned in my cs, Khthon doesn't really [i]need[/i] more valuables (though he certainly likes them anyways! Mortal craftsmanship is surprisingly lovely). A peasant uttering a sincere prayer could be more than enough for him, while he'd expect a king to make some more efforts. The more you give, the more you will receive in exchange.

For physical offerings, he has a particular fondness for gems and metals, especially when Mortals have crafted them into intricate tools, coins or jewelry, as well as stone sculptures, glass trinkets, fossils, and bone ornaments. Precious things that do not decay. He does not care for food and drink, nor for live sacrifices. He prefers his prayers and ceremonies to be silent, rather than loud or boisterous.

Some examples: Miners praying to find ore before going down in the mine for their shift would be a proper offering, and so would be burying some jewelry or small sculptures with a deceased loved one and asking for their grave to be protected. Explorers going spelunking can leave some of their wealth in the caverns they explore to ask for protection while within Khthon's domain, and someone trying to hide evidence (or bury treasure for safekeeping) could bury some coins along what needs to disappear so that it never surfaces again. As long as the physical offerings are placed within his domain (so underground in some way) then he can receive them.

If no offerings are made, then no boon will be granted: hidden things can be discovered, no protection will be given, ore will remain hidden, and buried treasures might be claimed by the God and never again be found by the one who hid it.

Taking more than his given is something done with intent. A miner mining the entirety of a gold vein he found in a cave or a tunnel would be fine, especially if he prayed first. If that gold vein was that easy to find, it was probably meant to be found. What is not fine would be doing stuff like aggressively blasting to try and extract just a bit more ore, grave robbing for the valuables that were buried with the dead (Khthon considers them within his domain, and so the valuables are his now), or desecrating and destroying his caverns, and any life that might dwell within them, for any reason at all (visiting and exploring is fine, just don't damage them). 

Above all, the selfishness of taking and destroying without asking first or giving back is what he abhors the most.

I hope that clarifies what I'm going for with the Secrets Domain! I feel like I might a gone a bit overboard haha
If you have any more questions, don't hesitate to ask ^_^