TL;DR: Dawnshards "bind" things to powers, sorta. They were used to make the Surge fabrials, and before that, were used on Ashyn to create the disease magic, a pandemic of which is what led to the Exodus.
I can't actually take full credit for the theory; I saw the theory they were used to make Soulcasters elsewhere, but I can't remember where to give credit, unfortunately. So far as I know, however, the latter theory is my own.
First, what do we know about these things?
One was in Aimia (and most likely still is, considering Brandon said "yes" before he caught himself). 
One is different from the rest 
Honor considered them important to the fight against Odium 
They were used to cause the cataclysm that forced both the Exodus  and the fleeing to the floating cities 
At least one was at one time believed to be able to "bind any creature voidish or mortal" 
Seem to match the Surge of Transformation much more directly than a normal fabrial
Use spren that are in Shadesmar, and potentially even Radiant spren 
Came from Aimia 
Other Surge fabrials:
Brandon seems to consider the Regrowth fabrial similar to Soulcasters in that it accesses a Surge directly 
The Ashynite cataclysm
Involved the Dawnshards 
Caused the Exodus 
"Forced people into the skies" in their floating cities 
May have been caused by disease magic getting out of control  (canonicity questionable, as it is from a reading from The Silence Divine)
Odium was "instrumental" 
Nahel bonds  and fabrials  are examples of Cosmere-wide phenomena
Essentially power of Connection 
Likely existed on Ashyn 
A Bondsmith "bound other Surges" 
Created or discovered the Nahel bond 
Magic used to destroy Ashyn used the "same basic principles" as Surgebinding, specifically referring to the Surges themselves 
Investiture usage on Ashyn was not always disease-based 
The diseases would likely still give powers off-planet 
Now, for wild speculation connecting these things!
Here's the ones I'm at least semi-confident in:
Soulcasters were made by the Aimian Dawnshard, binding spren to the device. It makes sense. They somehow have a spren bound to the fabrial, Dawnshards can "bind any creature", and they come from the only place with a confirmed Dawnshard.
A pandemic of one or more of the diseases granting magic caused the Ashynite cataclysms. In reading from The Silence Divine, the main character thinks about how staying away from Incubators is important, and how the "sky" (actually, destroyed ground) above is a reminder of the consequences of not doing so. While the sample is not necessarily canon, the theory does make sense. After all, when thousands to hundreds of thousands of people suddenly get crazy magic powers, and it's super easy to get them, things will go badly.
The Exodus was only of people who had avoided getting sick, or had gotten over any diseases they had had. WoB implies that the diseases could work and spread offworld. I don't think they'd risk carrying it over, and if they had, we would likely have some hints of it still.
The old Ashynite magic system was used to create the Dawnshards in the first place. Otherwise, where would they come from?
The one that is "different" is possibly just because the others are broken and that one still functions.
But what about the parts you aren't confident in?
Alright then, he's some much more stretchy parts of the theory/smaller separate theories.
Dawnshards were used to create, or at least enhance, the disease magic. After all, they were used on Ashyn, which so far as we know, has no spren. And the various spren bond types work with other things throughout the Cosmere. Why were the Dawnshards used to do this? Beats me. Science, I suppose. But the disease magics being created in a lab or something, and then escaping and nobody being prepared, makes more sense than the disease magic already being around, or just happening to occur when Odium was around. This also explains how the Dawnshards were so involved in Ashyn's destruction.
Some more potential evidence from a new WoB: the disease magic is from bonds between microorganisms and "spren-like Investiture" . If the Dawnshards can bind spren to things, perhaps they can bond these "spren-like" things to the microorganisms.
An Ashynite Bondsmith made the Dawnshards. Power over Connection seems like a good way to make things that bind things.
The old Ashyn magic was of Odium. Not saying every person on Ashyn was, but we know he was "instrumental" in what occurred, and that the planet had a full, proper magic system, which tends to only occur on planets with Shards. Unless there was another Shard in the system, Odium seems most likely. That system no longer being around is probably because Odium left.
An update to this: the "spren-like Investiture" involved in the disease magic could mean Ashyn has or had an equivalent to spren, and perhaps an equivalent to truespren. So, Ashynite Nahel bond for the old magic?
Each Dawnshard could bind one Surge. This one is even more speculative than the last ones. But the fact that Soulcasters are so common, and other Surge fabrials essentially unheard of, leads me to believe that perhaps the Dawnshards capable of producing the other Surge fabrials were destroyed or damaged. This leaves the "different" one as the one that still functions, and the Soulcasters as so common because they continued to be manufactured for far longer.
Either nine or ten Dawnshards, just because that's how these things tend to go. Maybe nine normal Dawnshards, and the "different" one is a tenth meta-Dawnshard used to make the others?
I'm here for crazy theories, these at least have some evidence or logic! I want my money back!
What, that wasn't crazy enough? It's a theory on Dawnshards, that basically makes it crazy speculation on its own! Also, what do you mean you want your money back, you didn't even pay me for this in the first place. Fine, here's some crazier ones that I don't for a moment believe, but momentarily popped into my head:
The first Nahel bonds were made using the Dawnshards. Syl says a Bondsmith created or discovered the bond. I have no idea what this could be referring to. And that's not sarcastic, I've got no storming clue. So maybe Dawnshards? But I don't really believe that. Maybe a Bondsmith was just the first Radiant, who knows.
The Honorblades were perhaps inspired by the Dawnshards, not necessarily in mechanics, but in spirit. This one, I'm more willing to believe, but I have no evidence for it, so I'm going with not the case.
The name "Surgebinding" could come because previously, they had to bind the Surges to things to use them. Now, they can just use the Surges themselves, but the name stuck around. Seems unnecessary though, Surgebinding just being because they bind the forces of nature to their will makes more sense.
[insert my usual disclaimer about typos and odd ways of phrasing things, because I'm too lazy for more than a cursory scan of what I wrote]
I'm not convinced of this theory, but I prefer it to just Surge enhancers. So yeah.
Also, anyone have any idea where the theory that Dawnshards made the Soulcasters came from? No idea where I saw it the first time.
And as usual, this theory is somewhat rushed and not super thought through. So may have some glaring issues I didn't notice, idk.
: The Way of Kings, chapter 75
: Oathbringer, chapter 113
: The Way of Kings, chapter 36 epigraph
: Rhythm of War, chapter 9
: Oathbringer interlude I-4
: Rhythm of War, Syl interlude from the newsletter