I’d thought about the possibility of a 4th runner faction as well. The ones that immediately jumped out to me were factions represented in the fiction but not necessarily in game - notably the NAPD and Opticon - but several of the important characters seem to be appearing on their own as resources for either the runners or corp in existing factions (Raymond Flint, Tallie Perrault, Caprice Nisei, etc.). I think an investigative faction would be most appropriate flavor-wise, as it differentiates them from the “art for art’s sake/underground celebrity” Shapers, “gimme that loot” Criminals, and “smash the system rabble rabble rabble” Anarchs.
(Of course, to be fair, even some of those stereotypes are breaking down - Reina clearly fights to dismantle the corps for… well, someone’s freedom anyway, as a freedom fighter - and The Professor isn’t so much an underground celebrity or an artiste tinkerer as he is someone with a good sense of morals fighting back in his own way. So there’s room for faction definitions to expand.)
Alternatively, there’s a number of small organizations in the fiction that fight the corporations strictly due to human workers being replaced by android (bioroid and clone) labor replacement solutions - Humanity Labor and the more radical Human First - although the latter would probably be represented by an Anarch runner.