In brief, it is a number between 7 and 12, generated at character creation. When something arbitrarily awful or wonderful might happen to you you may roll equal to or under it on 2d6. Much like saves in D&D. The difference here, other than there being only one "save", is that it is reduced by 1 every time it is rolled, successful or not and is entirely optional. The player can decide that he'll just take whatever is coming his way and save his luck for later.
Smart. Your Luck runs out. It's named something that you can even speak about out loud without breaking the tone. "Make a saving throw" or "Test you luck"? One of those is obviously stronger.
In the gamebooks from which the system is derived you would occasionally regain luck when choosing a path that, through no fault of your own, brought you fortune. Essentially you gained luck when you got lucky. This did not translate well into an RPG. In a gamebook there is no ambiguity, no fiat, whereas an RPG has a GM handing out the brownie points. A GM currency that keeps the PC alive encourages an irritating atmosphere of performing for the GM, "I did this thing, do I get a point?" or worse, conscious or unconscious favouritism.
The easy path would be to just give it all back when you rest, waking up nice and lucky. But strategic rests are also annoying, and it doesn't fit as nicely as regaining Stamina from naps does. There needs to be a concrete condition for regaining luck. I've already covered the possibility of regaining it through spending time with your family or getting wasted, but this isn't useful for long periods in the wilderness.
- You regain 1 luck every time you interact favourably with a suspicious aspect of the wilderness. For example, the party comes across a gauntlet on a pedestal in the middle of a room. Instead of wisely ignoring it or just stuffing it in a bag, the barbarian thrusts his arm inside. It's just a nice gauntlet. +1 Luck. This mimics the books quite well, is relatively plain and understandable. Encourages players to be a bit bold. Should also allow you to go over starting Luck.
- You regain 1 Luck every time you have a meal. Eating is good for mind and body after all. If you're stuck in a shitty jungle, hounded by pygmy cannibals, a nice quiet meal with your friends can restore some sanity. Further reward for eating and wasting time. Requires thought and risk.
- A good rest regains d3 Luck. Good dreams are important. Also, random encounters.
- Fake-out saves can't be used in Fighting Fantasy. It's incredibly unfair to spend their resources for nothing. Alternative: Use fake-out saves, but if the player takes the bait (remember, testing your luck is optional) give them 1 Luck instead.
- Regain d6 Luck at the start of a session. Represents the optimistic nihilism of the adventurer.
A lot of of these come down to calling the GM's bluff, which I think is appropriate. Rewarding curiosity is never a bad thing.