Key already used

wrong category
and since he uses this website my guess is that he read this already

I made a mistake in a giveaway that i created.

You're doing it wrong.

I didn't knew that Humble Bundle Giveaway keys had a deadline, and someone won an expired key.

Humble keys don't expire, they only disappear from your account. If you took note of it, it should still work. Of course, if you didn't reveal and save the key somewhere else, you've lost it.

I suggested giving him a key to another game (i heard that support can change the game int he giveaway)

You're doing it wrong. The winner can't accept another game because it is against the rules and he could be suspended.

It was the second time that i made a giveaway, and the first from a Humble Bundle Giveaway.

Yet you made a third one for "The Red Solstice", which was also given away on Humble.

I accept that it's entirely my fault, and i want to find a way to apologise to the winner.

You're doing it wrong.

I'm guessing the giveaway you're referring to is the one for "Company of Heroes 2". I'm sure you can easily find a working key on steamtrades that you can give to the winner. You could also try to convince him to remove that giveaway, but that's up to the winner.

Thank you.

You're welcome!

Yeah, you just need to talk with them. If you can't get the GA changed, then delete it (with the winner's permission). Friend the person on Steam so that you can chat, and just come up with a fair solution. When I've messed something up, I've found 5-6 games in the same price category in my list of keys and then just given the winner their choice of all of those. I've only had to do this a couple of times, but in each case the winner was perfectly happy with the solution (I think that in both cases, they actually probably got a better game than what they won).

changing the giveaway game isn't that easy.

just get a new key for the winner or ask him to delete it.