New Badges: At least 1, surely. A lot of remakes have at least 1 new item, so a new badge seems par for the course.
Luigi Side Mode: As much as I pray for this to be a thing, I doubt it will be. Too much work. Sure, Pokemon got DLCs with their own story, but that's a mainline game for #1 grossing media franchise & I could believe they planned DLC from the start. TTYD Remake is a remake. I doubt they put in the planning or budget for whole new stories as DLC. : ( I'd be surprised to get new items.
Sound Test: Sticker Star got one, & I think SMRPG remake had it, too, so yes.
Postgame Boss Rematches: SMRPG remake did it, maybe? It could be used for hype in ads with not much more than small cosmetic changes, bigger numbers, & maybe a few new mechanics.
Item Storage: Maybe? 32 items storage was probably chosen because 32 bits or something. Changing that might be more awkward than the remake devs prefer.
Vivian is Trans in English: Saying Vivian's transgender specifically seems like it'd be a commitment in terms of how much text it'd take. I'd assume they'd just use female pronouns for Vivian.
Gambling references removed from Pianta Parlour: These are likely to be removed because of EU anti-gambling laws, which I've read supposedly recently outlawed referencing gambling, too -Which'll make the Toad ruining all her coins by gambling awkward- so these seem likely, at least for European versions, & if they're gonna make it different for the EU version, why take the effort of having it different for other versions?
Fast Travel System: Maybe? Just a loading zone change, so it'd be easy, & SPM had the Return Pipe, so....
2nd Pit of 100 Trials: Remake difficulty is interesting, but Intelligent Systems ain't Square Enix. I'd be surprised if they cared enough to put in a 2nd Superdungeon.
Original Music Toggle: Presumably almost as simple as a sound test, & SMRPG Remake did that, too, so this seems plausible.
Hard Mode: Devs going to the effort or revising dozens of enemy stats, & potentially even AI & badges & costs & item effects & stuff? IDK, seems like too much work....
Prerendered Cutscenes: Maybe? They didn't show any in the trailer we've got so far, & something like that seems big to omit in the big 1st promo for something that looks so finished so far. What could be at the right level of importance?
D_______ naming uses NS keyboard UI with "p" missing: Either they remake a whole UI/AI/controls for a small joke, or remake the cutscene to add a small joke/forced failure even if you can use the "p" via the NS keyboard.
Boss Rush: Potentially? My stance on this is the same as Postgame Boss Rematches; 1 is putting a boss rematch somewhere, the other is setting them up to happen one after another. Doesn't seem that complex.
No Fog Stage Hazard: It's too well-known & iconic to remove. If they're purist, they'll keep it in. I could be projecting, though, because I'll be genuinely upset if Fog is removed.
Museum with Concept Art: They still have access to the concept art?? Making a whole new location for that? Has IS done such things before? Maybe in Sticker Star onwards or such?
More New NPCs like Glitzville Purple Toad: If I'm interpreting this, it seems like there's precedent. It'd just be some lines of text, collision & wandering, right?
Better Base Item Inventory: Why would they change this? & if you get 20 item limit without the Strange Sack or such, what replaces it?
Japanese Title Screen Music in all versions: I'll be upset if they do this. In part because the JP title screen music is subtly, kinda disconcertingly different from the music in other versions.
New Moves for Partners: A purist remake adding new moves, thus changing the whole balance of the game's beloved battle system? Doubtful.
Slightly Different Blocking/Superguard Timing: I could believe this would happen just because of hardwar/software differences or awkwardness in remaking the timings, or they just have stuff they wanted to change.
Trouble Center Improved: Hopefully. I wouldn't be surprised if they added at least 1 thing for remake.
Amiibo Support: I don't know of many instances of PM games getting Amiibo Support. Why now?