How can I change what faction I am "Closest To"?

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.)