I would LOVE to see Once On This Island do One Small Girl into Waiting For Life, which conveniently ends with bits of One Small Girl anyway. Together, that would set up the show so well for people who don't know it.
I think so much of this depends on who's actually nominated. Like, If Kenita R. Miller & Phillip Boykin are both nominated (I assume Hailey Kilgore is a shoo-in for a nom), this would be a solid choice. We Dance would be perfect too.
But... they'll probably do Mama Will Provide. I've made my feelings on Alex Newell clear in other threads so won't write an essay here, but man will I be incredibly disappointed if they make that choice. (In a nutshell, that song gets wild applause by default in literally every production of OOTI - because it's a rousing, sassy production number - but I think his is one of the weakest I've seen. I also think the beginning sound-making part comes off as cringey children's theatre when it's out of context.) It's been one of my favorite shows for the last ~25 years, so selfishly I just want a Tony performance that I'll want to replay over and over. Any other song would fit that bill!
Otherwise, I kind of hope The Band's Visit does Welcome To Nowhere into maybe Omar Sharif, though I agree with others that Answer Me is stunning, so would be very content if they made that choice. I do think Welcome To Nowhere sells the show really well to people who don't know what the hell it's about. There's so much wit and humor in the production, and I don't think Answer Me shows that off, as beautiful as the song is. (I also think they'll probably want to show off Katrina Lenk as a frontrunner for Leading Actress.) I keep thinking back on Groundhog Day, and how Seeing You is beautiful, and so poignant in the context of the show, but fell totally flat out of context at the Tonys and did nothing to show what the show actually was. That's different than The Band's Visit which obviously has multiple poignant moments, but still.
I think Frozen will depend entirely on the nominations! But I have to think they'll do one of the original songs from the movie... probably For the First Time In Forever. (Though I wouldn't argue with them going all in and doing Let It Go. It certainly would sell tickets. Don't know the tech details of that costume change moment and if it's possible at Radio City, though.)
Do we think Harry Potter will perform something? I don't see it until late June so can't personally guess, but I keep hearing how beautifully "choreographed" it is.
Haven't seen the other shows yet so it would be useless for me to guess based on hearsay, though I do think June Is Busting Out All Over will make its way in there somewhere, if not as Carousel's official performance than somewhere in the opening. (Can we get a Leslie Uggams cameo??)