What are the strengths and weaknesses of the factions in the Call of Cthulhu LCG?
I've done a bit of my own research, so I can provide what I've learnt so far. It's worth mentioning that this is not what I've learnt from extensive play, but rather what I've gleaned from comments and articles about the factions.

Agency - Quite straightforward strength in combat, plus some have Willpower which offers resistance to being driven insane. Supposedly they're well balanced and a good "support" faction.

Syndicate - Sneaky. Lower play costs allow them to be played early, plus the ability to manipulate other characters (via exhausting and reducing stats) and stories (via switching) make them hard to pin down. Decent at combat but vulnerable to insanity.

Miskatonic U - Tons of investigation, plus decent arcana, but pretty awful combat and terror effectiveness. Also has decent deck control (drawing and filtering) and can sometimes manipulate the struggles to their advantage.

Cthulhu - Very powerful combination of combat and terror. The downside is that it's an expensive faction to play so takes a while to get moving. Card destruction is the big theme within the faction.

Hastur - Terror (plus lots of non-struggle insanity) and hand manipulation, as well as the ability to prevent effects are the strength of this faction. Weaknesses include the inability to actually destroy other cards, and a shortage of decent characters.

Yog-Sothoth - Card filtering and discard pile manipulation, but with some weakness to insanity. No particularly great characters, but they'll keep coming back.

Shub-Niggurath - Acceleration and characters that improve each other when they're all on the board together. Decent combat and terror, but lacking in arcana and investigation and not particularly good at removing opponent's characters.

Hopefully these notes can give a decent starting point.