Option #1: Oblivion in laudanum.
The solution is extremely Victorian: F.F. Gebrandt's Superior Laudanum, to help you sleep at night. You can find it in the Echo Bazaar (click the "Bazaar" tab), in the shop Nikolas Pawnbrokers. Each bottle costs one Echo, which can be quite pricey if you're trying to sell Whispered Secrets or the like in the Bazaar.
Drinking laudanum is bad for your health, and your Wounds quality will increase a bit one change point), but you can reduce that with F.F. Gebrandt's Tincture of Vigour. I imagine that Gebrandt is an extremely wealthy lady. Alternatively, you can just rest at your Lodgings if your Wounds quality is 3 or higher.
Once you become a Person of Some Importance (PoSI), as with the Tincture of Vigour and Ablution Solution, this option will become less viable; unlike the other two, it will also have a nasty side-affect: every time you drink a bottle of the stuff, you'll become slightly more addicted to it, resulting in the Laudanum becoming less effective.
Option #2: We're all mad here.
Go to A state of some confusion. It will discard and replace all of your Opportunities with new cards, so don't do it if you really like your current Opportunities. Almost all of the Opportunities and actions available in confusion will decrease your Nightmares (though some will abuse your base stats a bit, and one or two will increase your Nightmares while giving you something else you want), and you will leave A state of some confusion once your Nightmares drops to zero.
There are two ways to get to A state of some confusion:
- Go to Mrs. Plenty's Carnival, into the House of Mirrors, and then Gaze Into Dream's Mirror, which requires that you have at least one in Nightmares.
- Let your Nightmares quality increase to 8. You'll be taken there immediately.
A state of some confusion doesn't cost you any Echoes, but has a much worse action-efficiency compared to drinking laudanum. (Laudanum seems to do about 4 change points of Nightmares reduction, and actions in a state of some confusion are more like 1-2 change points of Nightmares reduction.) Not all actions there will decrease your Nightmares, but drinking laudanum will waste some of your time with curing your Wounds, so it's a wash; there's some good reasons to just let yourself go mad. Unfortunately, a few of the actions in confusion have a chance to abuse your four main qualities, and there are a few other small-ish drawbacks, so it's not all upside.
Option #2 if you have Memories of Light: we're all mad here, too.
If you have 10 Memories of Light, then you can spent two of them to go to the Mirror-Marches instead of A state of some confusion at the carnival. If you have at least one Memory of Light, then getting to 8 in Nightmares will take you here instead. The Mirror-Marches are much more generous than A state of some confusion; your main qualities won't be abused nearly as much, and actions that remove two or more points of Nightmares are much more common.
Option #3: Psychotherapy.
In your Lodgings, you'll get an option to Find a way to deal with your Nightmares if your Nightmares quality is 4 or higher. Spend a Sudden Insight (possibly gained via Invite a friend to join you in something terribly intellectual from your Lodgings), and you'll lose 5 change points of Nightmares. Your friend will gain one change point of Nightmares and also spend one action, so you should return the favor sooner or later.
Lacking lesser Rumour items, you can use the Implacable Detective's Business Card to exchange Whispered Secrets for Incendiary Gossip, if you have it.
More Opportunities to acquire Incendiary Gossip can be found by cultivating your connections with Bohemians, the Church and the Great Game.
Note: There is a list of Storylets that result in acquiring Incendiary Gossip, if you wanted to see whether any Opportunities you currently have available would result in Incendiary Gossip (but that would be cheating now wouldn't it ;) ).
Best Answer
About January 14th, 2014, there was a change to the faction cards: each faction card now has a storylet to abandon that faction (losing your Closest To quality entirely, along with every single bit of your Connected: [that faction]), and a second storylet to become Closest To that faction if you're not close to anybody.
Becoming Closest To a faction is quite expensive: it usually costs several of an item that you can only acquire through the Bazaar Sidestreets and/or the Relickers – for instance, becoming Closest To: Hell requires 5 Portfolios of Souls, and becoming Closest To: Tomb-Colonists requires two Parabola-Linen. Those high-level items take around 160 Certifiable Scrap per unit, so becoming a Person of Some Importance and visiting the Bazaar Sidestreets is usually easier than acquiring a truly heroic amount of Certifiable Scrap. (Even if you're not yet a Person of Some Importance, it's still easier to become one than it is to acquire the 300-plus Scrap!)
At the moment, the faction that you're Closest To only affects your ability to participate in the various Counting the Days storylets, which are mostly on opportunity cards. These are more-or-less balanced across the factions, with two striking exceptions: both the Great Game and the Bohemians have access to both of the always-available storylets Dice games on the quayside and Seeking Curios and Secrets in the Forgotten Quarter that will raise your Counting the Days quality if it's in a certain range. The "low-class" factions can only access the first storylet (taking Counting the Days from 0 to 6), and the "high-class" factions can only access the latter storylet (taking Counting the Days from 6 to 10). This means that getting up to Counting the Days 10 is fast and easy if you're Closest To the Bohemians or the Great Game.
The other way to change your Closest to To is to sell the location of the Cave of the Nadir to one of four factions: Hell, the Urchins, the Great Game, or the Revolutionaries. This will give you a pile of Connected: [faction], a highly-valuable item, and will change your Closest To to the chosen faction. (Finding and completing the Cave of the Nadir is a long storyline outside the scope of this answer.)