I am looking for a way to use some sort of luck points in my grim world campaign. My players seem to have really bad luck and I often feel sorry for them, when they fail after fail. As an example during a fight against a lesser demon one player has rolled 6- three times in a row and got quite pissed off for the remaining game time. I think it could also be an interesting way to reward players for being active and great roleplaying.
A 6- doesn't mean the PC fails. It means you can do whatever you like.
Of course, if you give them a better outcome than they'd get on a 10, your players are going to cry foul and I won't blame them.
Your blow cleanly severs the demon lord's arm. It thrashes about where it lies, then launches inerrantly at you and grabs you by the throat!
is a pretty neat thing to do on a 6 and still progresses the fight. It progresses it toward something worse, especially if you turn toward whoever's next and talk about the new armored limb with spikes on the demon lord grows, but it's much less likely to make the player think that they "missed their turn".
But even given that, a prolonged run of bad luck can still grate, so here's a summary of how to use the luck token system, detailed on pp. 156-157 of Grim World.
Each character has three weaknesses. Describe them. Give them tags like the animal companion weaknesses, and be brief. Proud. Stubborn. Scarred. Fearful. Stuff like that.
When your weakness might apply to a situation, you can volunteer how it might make things worse somehow. If the GM agrees, take 1 luck token.
At any time after you roll a die or dice and don't like the result, you can spend a luck token to reroll one die. This includes the dice rolled for things damaging your character, which are your responsibility even if the GM wants to use their big fancy dice bag.
When you need a lucky break, you can spend a luck token and the GM will drop something good in your path, however improbable that might be. Take +1 forward to take advantage of it.
Those are the official extents, but there's something I like to do running Fate games, which might apply here since the luck token mechanic is written as kind of a Fate/Dungeon World hybrid since the setting itself switch-hit for both systems. If someone rolls something egregiously bad and that fits with one of their weaknesses, rather than spending a luck token, offer a luck token along with a description of how their weaknesses contributed to that terrible outcome as a bribe to let the roll stand.
This is not a good idea for Dungeon World and sounds out of place for Grim World.
Failures have fictional consequences. A handful of moves have specific responses to a 6-, but most don't and fall to you, the GM, to provide. Failure is not without its own rewards, since the players get XP along with whatever new twist you've provided. Yes, players might feel down about their worsened situation, but if they survive it, the XP is there for Leveling Up afterwards. Grim World, more so than Dungeon World, also places an emphasis on character death with world-changing Death Moves.
Embrace the instruction and advice in the GM chapter of Dungeon World, and remind players of the power and story possibilities of a good character death in Grim World. And keep in mind that two of the answers to Why play Dungeon World? include see the characters do amazing things and struggle together.