|Pictured: Not a hat|
What We Know
Ignoring the flannel about the sex/race/orientation/favourite colour of the characters, what we now know is that the main character isn't a Captain but a Lt. Commander, that the show despite being named after one ship apparently features multiple crews, and that it's set 10 years before TOS. (First note for Trek geeks- this means the TOS Enterprise is already out there, commanded by Christopher Pike, since Kirk's mission starts in 2265 and Pike had her in the 2250's) The ship is of a design we've not seen before, looking a bit Klingon, is based in an asteroid (apparently) and has the ident code NCC-1031.
What it might mean
Hard HATS on!
So, this class of ship is about in the TOS time period, but we've never seen or heard of it before. Its registry code contains the number 31, which several fans have taken to be a reference to Section 31, Starfleet's shady Black Ops division which first popped up in DS9 but have since been seen in Enterprise and ST: Into Darkness. Putting this all together, if the ship were controlled by Section 31, and secret from the rest of Starfleet, this might explain why we've never heard of her class. Of course, it would help if no-one else could see the thing- for example, if it had a cloaking device. The Klingon design elements might point that way, though it was the Romulans who introduced the technology in TOS. Nonetheless, the NX-01 crew had contact with versions of the technology so it might well be that Section 31 have access to it and hid it from the rest of Starfleet.
What about our new lead character, then? Well, if the Discovery is a super-secret cloaked ship run by Section 31, she might well be a character whose role is to be embedded in the crews of other ships on secret missions- maybe 'helping' the Starfleet crews in ways the Prime Directive and other regs might not allow, or even sabotaging missions that her bosses don't want to succeed. Initially, I'd expect her to be simply a spy, tasked with keeping an eye on occurrences on other ships (maybe even non-Federation ones), which would obviously tie in to the name of the ship/show nicely. The Discovery crew would then become her support staff, making the show basically a Star Trek version of Alias.
By the way, all this means they could potentially, in a later episode, have our infiltrator on the Enterprise during a TOS episode, Trials and Tribble-ations style...
I think this is a fun little HAT, ticking the boxes of being interesting, fitting the known facts, and probably being utterly wrong. Regardless, I'll be watching the thing, even if they get Bieber in to do the intro music. I mean, they won't do that, of course.
Please don't let that be the only thing I'm right about.