There is no equivalent to this group in DC's universe. Jonathan Hickman's run on New Avengers gave readers a snapshot of what a DC version might look like but... I'm not convinced by the group he assembled, if only because it had a Superman equivalent among its ranks, and the secretive nature of the Illuminati seems at odds with the Man of Steel's way of doing things. If it were up to me, I wouldn't put him in there at all...
So who should be there?
Let's start with Batman. He's an everyman (sort of), the best detective on the planet, and not above going off the books to get things done. He's a link to the JLA and provides a good lot of brain power. Also, this is the guy who comes up with plans within plans to fight villains and has contingencies for if his friends go off the rails...
Next up, Queen Hippolyta of the Amazons, leading one of Earth's most advanced and secretive groups she would be ideally placed to resume her Wonder Woman role (from World War 2) .
Another Queen gets added next, Queen Mera of Atlantis, pulling strings behind her husband's back and running with dark and dirty types to maintain his kingdom might well be her style.
Green Lantern... by which I mean Alan Scott should get a seat too. Aside from being a powerhouse, he's another link to the Justice Society and represents the older heroes better than Hippolyta.
Amanda Waller... yes that Waller, the one from the Suicide Squad and the DC's Queen Bitch. She has to be at this table, if only because she's hugely influential behind the scenes.
Lastly (because it would be wrong to go over six members), we need a mystic, and I think that means Dr. Fate, as the DC Universe's answer to Dr. Strange.
The only other character I would add would be the Martian Manhunter, because of his link to space.