I'm not exactly breaking any ground by saying Lucky Cigarette Case III is good. But I did want to take a little bit of time to write about what has proven to be a rather dominant card. I've noticed that at my table just about every Rogue deck takes this card. And people are choosing to play Rogues just because of this card. It's that good.
Lucky Cigarette Case III does not need a lot of support to be awesome. You don't need to be a dedicated "oversucceed deck" to make this card great. Passing skill tests by 1 or more is something most investigators will routinely do almost every round. This will draw a lot of cards, with the upside of a fair amount of selection, for just about anyone. Of course, if you are playing oversucceed enablers, like Lockpicks (and why not? These are some of the best Rogue cards in any case), all the better. You can find some extra synergy for the Lucky Cigarette Case here and there among the Rogue class (Trish with her Myriad cards, Winifred with her natural emphasis on oversucceeding, Parallel Skids with his free skill test every turn), but really, this card is good for just about everybody. A Preston deck aiming to avoid the chaos bag probably doesn't want this, but in my opinion that's really about it.
A particularly common setup is to use Relic Hunter to get two Lucky Cigarette Case IIIs in play. It's a truism in card games that a great strength of card draw is that it draws you into yet more card draw, and so on. (This is probably even more true in Arkham than other comparable games because here the penalty for drawing through your whole deck is so minor.) So, you'll find that once you have two Lucky Cigarette Case IIIs in your deck, you're much more likely to get one into play early, and once that's in play, you can use to to quickly dig out the second. Then you'll just fly through your deck.
What can you do with all this card draw and selection? Well, whatever you want, really. The sky is the limit. I'll say that there's particular synergy here with the Myriad cards like Three Aces, Astounding Revelation, Segment of Onyx, and so on. Also, once you have a ton of draw, be on the lookout for cards that are useful but cheap in resources and actions. Cards like Shortcut or Quick Thinking, where the primary cost is the card itself, look quite appealing when you draw tons of cards every turn. Also consider ways to efficiently turn a surplus of cards into money--stuff like Cryptic Writings and Easy Mark. I'll also mention Dream-Enhancing Serum. Obviously, when you have tons of selection from a couple Lucky Cigarette Cases, it's not going to be hard to get yet another card out of the Serum every turn. And when you have that much draw, the extra hand size will matter.
Also, don't be afraid to intentionally draw weaknesses with this card. Some investigators, like Trish, have weaknesses that are very mild if drawn in just the right situation. So if the opportunity arises, it can be a nice play to cheaply dispose of a weakness like that.
So, a really great card, almost regardless of what your deck is doing. In fact, as far as my group is concerned, the other Rogue accessories might as well not exist. Is that a healthy balance? Perhaps not. But it's nice to have a card that makes people want to play Rogues, and this certainly qualifies.