blockchain – Can you potentially mine for anything?

This is more of a general “blockchain” question and not specific to Bitcoin so if I’m off topic here I would appreciate it if you could direct me to a more relevant stackexchange maybe.

On PoW-based blockchains, participants mine for unique hashes with a specific number of leading zeros etc. (there’s probably more to it but it’s irrelevant to my question).

But I’m wondering whether a blockchain could ask participants to mine for something else. Think of it as a game, a challenge. Let’s say a string of characters (bytes, a file whatever..) “hidden” inside the blockchain. Mining for that string should (as in: that’s the challenge) be as hard as mining for these unique hashes.

My questions are the following:

  1. Is it possible?
  2. How/Where would you hide it (think of a “custom”/theoretical blockchain, not any of the existing ones necessarily)
  3. How would you allow users to mine for it without giving too much away thus making it easy for them to hack their way around the mining?

When it comes to mining for unique hashes there is nothing to hide because you’re trying to generate a random sequence of characters that doesn’t already “exist” somewhere. But in my hypothesis, the “hash” needs to necessarily pre-exist “somewhere” but the user shouldn’t have access to it, only “tools”/hints to mine for it..

Thank you in advance for your help 🙂

I’m not a blockchain expert (I am a front-end Js developer) so I’m looking for more “high-level” answers 🙂