Since there may be those here with fertile imaginations (or opinions on the froth rising in the ferment), I thought an update might be in order.
Here are the characters who've been brought up so far, one way or another.
We've had two suggestions for his faction: the Bourbons (one vote), and the Musketeers (three votes). It has also been suggested that the Musketeers would be a good name for Vachon/Screed/Bourbon as a trio. Or alternatively, the Lost Boys for the whole of Vachon's crew, however many they might have been at one time or another.
In the Season Three Production Notes, Vachon's group were referred to as the "Lost Boys". Several were described, including Bourbon, who is supposed to have been a Musketeer in France in the 16th century (and hence probably minor nobility) before being brought across. However, none of this ever actually got used in any of the episodes.
EDIT (30 June 2012): It turns out that there IS a faction for Vachon's crew. Called (not inappropriately) The Crew. It follows that "the Musketeers" is wholly available as a name for Bourbon's faction. Also, of course, it follows that there will shortly be a Crew icon in the works.
So far no one has objected to the idea of repurposing the Vetterans (an early name for Tracy's faction) to name a faction for Barbara and Richard Vetter, and Tracy's relationship with them.
So far, two names have been proposed for a faction for Divia's mother, Seline: the Selenoids (suggested by Andrew) or the Lunatics (suggested by PJ).
Most of the discussion has actually focused on the spelling of her name (Seline seems to be the one used in the credits), and whether she was the madam of a brothel (which is definitely what the script says).
So far only one name has been suggested for a faction for Emily: the Denied (suggested by Roger). At this point, three people have said they like it.
Joan of Arc:
At this point, three names have been suggested: the Messengers (Roger), the Arcatypes (Andrew), and the Arc-en-Ciels (me).
Two suggestions: the Hackers (Roger), and the Sorcerers (Andrew).
Chief Inspector Liam O'Neal:
Apparently the Celtic Glow-Worms had a love/hate relationship with him that they called the Celtic Connection. However, he doesn't have an actual faction of his own that I know of.
I've just suggested the Irish Charms, but other ideas may be forthcoming.
Others for whom names have been suggested:
Most of these were tossed out by me, but not all—and so far they've not been commented on, so maybe they didn't hit any of the right spots.
Ellen/Monica/Jacqueline: the Triple Threats (Roger)
Qa'ra: Ancients, Ancient Ones, Qa'ranians
Serena: the UnSerenes, Serenities, Cross-Dressers
Francesca: Francescans, Reincarnates
Alyce Hunter: Alycians, Archaeologists
Pierre Rochefort: the Priests, the Confessors
Characters for whom no names have been suggested so far:
Angel, the vampire who brought Vachon & the Inca across
Mai Tung Wa, the Chinese acupuncturist
Patrick McDonagh (or his father, Robert)
Norma Alves, the computer tech at the 27th precinct
Eddie, the morgue attendant
In fact, I seem to recall someone wondering whether there was a faction for the duck lamp. (Or was it the moose pyjamas?)