You need to hold the coins for (if I recall correctly) 8.8 days before they are eligible to stake. After that, it's just time with the wallet open for staking. Also, walletpassphrase is a different function. I don't have HYP on this computer, it's miles away

but there's a button, I think under "options", that you need to push to unlock for staking. walletpassphrase MIGHT work, but I think not. It's been a while. My HYP stash is pretty minimal, would probably take more than a year to stake. I would expect at 4K coins, a week or more. The more you have, the faster you'll stake.
The command walletpassphrase will allow for staking, but only for the fixed amount of time AND all other commands are also unlocked. If you unlock the wallet for minting only you can't then send coins to another address without entering your passphrase. It's an added level of security just in case someone can remote in or has physical access to your machine.
Well I think that's not complete info at all. At first (in my case) I run the headless (non-gui) wallet. So there's no Option-Menu ;-)
And second it's not completly that walletpassphrase is "unsecure" at all. In fact you can (and must) not only limit time it is running unlocked, but you can also additionally add "true" parameter to use it for mint (and also staking?) only.
This is said by CLI/RPC info:
walletpassphrase <passphrase> <timeout> [mintonly]
Well I'm not completly shure if this is a misinfo/typo for "mintonly". Does this include staking? How else should I be able to activate this on headless/non-gui version? I've also added "staking=1" to my conf-file.
But I should also have said I'm still waiting for rewards on my 4221.000 HYP-Coins deposit for testing having had not even a satoshi generated yet

My apologies, I assumed you were running QT. Yes, you're correct, that is how to set it for minting via command line. and yes, mintonly is the correct parameter.
Running that way, assuming your coins have reached the maturity date, it's just a matter of time. Eventually, they will stake. The more you have in a single transaction, the faster it will work. 4221 is actually a really small amount on the current network. I don't know how long that would take, but I suspect rather a long time. Over a year ago, the suggested minimum was 20K, and that was gonna work (then) about a week after eligibility.
If staking is your primary goal, I would highly suggest acquiring more coin. I've been out of the loop too long to even suggest a number. I'm currently building back up to where I think it will work out, and I'm aiming for a minimum of 100K coins.