If the bitcoins were lost for some reason, no one will find them. They completely disappear from circulation. This is the essence of the digital currency. Therefore, it is necessary to reconcile that many bitcoins will disappear forever, if someone has forgotten the password from his mobile wallet or for other reasons has ceased to have access to it.

Brute forcing private keys is hopeless. What might be possible is recovering bitcoin from wallets whose passwords have been lost if the wallet file is retained, if the passwords are not too difficult to crack. And there might be bitcoins considered "lost" because the original owner has died, and whomever inherited their equipment has not noticed/found/deciphered their bitcoin wallet setup (or paper wallet tucked away somewhere). There are varying degrees of "lostness" to bitcoins that are considered lost.

A digital currency that has been lost due to the forgetfulness or death of a user, a broken technique or intentional destruction can never be restored.
In addition, several thousand bitcoins were deliberately destroyed. For example, the start-up Counterparty "burned" 2130 bitcoins in the process of distributing its own digital currency called XCP and it was not a single case.

The only way to "find" them would be to stumble upon the private key that holds them.

The odds of this happening are so small that it is basically impossible to find them.

What's lost is lost

But there can still be a workaround, though. Let's assume that Bitcoin doesn't die in a few hundred years from now on, and our descendants (or even us ourselves if we manage to live that long) could reclaim these coins (coins that didn't move all that time) as public property for the common good (after some grace period expires within which you could claim the legitimate ownership of them, of course). It would require some modification to the Bitcoin protocol, but other than that (and consensus required for that), there are no particular hurdles or issues to do exactly that, i.e. find them back

Could a bitcoin mining rig be put to use try to hash the right private key - or is that a stupid answer?

They are burned bitcoins and there's no way for anybody to recover them even Satoshi himself, unless he'll do something with the codings and will recover the private keys and the seeds of this wallets. But if bitcoins are gone now, they are totally lost forever. And that's making the supply lesser in the market others are burning bitcoins.

"Lost coins only make everyone else’s coins worth slightly more. Think of it as a donation to everyone" - Satoshi Nakamoto

As I've been saying to my other posts, bitcoin is not really anonymous, there is been a proof about that. The reason that lost bitcoin can't be found is because paper wallets need your passcode, but if you save it to your computer and deleted it, you're done. Also, if you send them to a wrong address which is not really a bitcoin address, that will be void.