Lady Secretik

Lyonel and his giant knight. In a normal world where Lyonel gets his cookie, Dunk can be his man and live at Storm's End, maybe he has a tolerant Fujoshi wife who is more than happy to side step the marital duties. But in general he cannot marry the man and he can't have heirs with him, as much as he desires he can't really declare Dunk to be his openly to the four winds.

BUT in an omegaverse world with omega Dunk, all those problems disappear, sure there is still the class difference but that's overcomable. Gender is irrelevant because you have men with cunts and wombs going about and children are a given with heats and ruts; there's a lot to work with, there are fated mates, there are mating bites, fusing your soul into one and all that jazz.

In conclusion, Lyonel can make Dunk unequivocally and irrevocably his for everyone to see. And (I saw this in a fic, thank you @Fayefaye90) since Lyonel himself fell in love , married and mated a low born omega, his rage against Duncan The Small is rather ironic. I don't think any rebellions for broken engagements would happen with Dunk on Lyonel's lap soothing his rage with memories of their own love. Maybe some restorations for sure, but no tragic trials by combat.

And into the next generations of Stags

Robert Baratheon following his (great?) grandpa's footsteps and falling in love with a noble, kind, strong omega. I can totally see Robert being a proud and a fan of his own family predilection of mates btw. I'm also assuming he is even bigger than in canon thanks to Dunk's genes and because I adore happy endings he is also less of a dick than in canon; the seed is strong but who says the womb isn't strong too. (at least some of Dunk's goodness has to be passed down too. )

I remember reading a fic where Ned didn't present until his late teens, after everyone assumed him to be a beta. And because of that he was being fostered at the Eyrie with Robert, learning sword, riding and all that.

We all know what happened in the canon. But in an omegaverse world, Robert can have his Stark. Especially the Stark he has a deep connection with and has loved ever since he was a child. I don't see a Ned who grew up under the tutelage of Jon Arryn and next to Robert ever leaving them for Rheagar. Hell, the war can be avoided all together if you move the pieces right. Or it can still happen but the 7 kingdoms would be a wholly different place if Robert is the king with Ned as his Queen.

Renly and Loras ? Stannis and Davos? Same thing, slap an omega tag on one of them and problem solved. Or maybe play with other designations, either way I adore how the omegaverse opens space for so many different possibilities and how many of them could end up with happy endings for my favorite house.