Okay, so I've thought about it for a bit and I think I've cracked it.
What we know is:
- Solrun's father, Runor, is responsible for Drysalt's initial rampage. Given his practice of Sedir, this strongly suggests some kind of summoning, although "rampage" suggests limited control at best.
- Hasvir Hadinghero kills Runor and ends Drysalt's first reign of terror through unknown means.
- Solrun is angry and wants revenge.
- Framarr Crownsword (Dorri's father), hires Blackhand to kill Hasvir for Solrun, in exchange for The Secret.
- He also compels her to swear a binding oath not to tell The Secret. (This is important; it implies he really does not want people knowing about the deal.)
- Solrun strongly believes that we will need The Secret if we are to have a chance of stopping Team Dorri-Drysalt. She seems to believe it's essential, worth her life.
So, what's The Secret?
The Secret is nothing more or less than the knowledge of how Drysalt is called or summoned, which necessarily also contains the essence (probably at great cost) of how to banish or repel Drysalt.
This fits with what we know happened. Drysalt was summoned by Runor, caused havoc, and was then
cast out rather than killed. (Either because Runor's death ended the spirit-pact that allowed Drysalt to remain, or because Hasvir or maybe even Runor worked out how to banish him.) It also makes sense why Framarr would have wanted The Secret - either as a nuclear option to use against his rivals, or maybe for altruistic ends, to be able to banish Drysalt again in case he ever returned.
The fact that Drysalt has not gone on a massive rampage again (yet) is most likely because Framarr was patient, and spent a lot of time working how to more reliably summon/contract with Drysalt, and passed this knowledge onto Dorri. Or because Drysalt is more patient this time, and is lulling them into believing they have him controlled whilst sock-puppeting them. It's effectively the same thing. The fact that Dorri approached Sten even shows that he considered other, less extreme options, before turning to The Secret.
It fits with why Solrun things we absolutely need this information or we have no chance. If we don't know how to banish Drysalt, then (in Solrun's mind at least), we have no hope of defeating Dorri. Even Corpsemaker may not.
@Imperial Fister, asking if I am right on the broad strokes here, for a reward dice if need be.