Everything Old Shall Become New Again
Voldemort is reborn as one of Harry's children, and they begin to slowly regain some memories during their Hogwarts years. Sort of a slow-burn redemption arc.
Optional: My own concept of this idea has the broken pieces of Voldemort's soul coming back together and healing while they forget what they've done, and then being reborn again slightly different because of the recombination. Harry's daughter Lily is born with a twin sister, who he and Ginny name Frederica, after Fred, and she starts remembering things from her past life after seeing some of Voldemort's old possessions in a museum. She then has an existential/self loathing crisis of sorts when she eventually figures out who she was.
Sleeper Agent (2.0)
I'm sort of already doing my own version of this (It should be the same title, you can find it on whatever site you're reading this) but I'm by no means claiming it for myself. I've condensed the general idea down a little more than when I wrote it down earlier in this document. The general idea is thus:
At some point during the second war with Voldemort (mine is an AU where the war went on for much longer than canon) Voldemort experiments with a spell that ties his mind to an unassuming Hogwarts student's, allowing him to control them from behind the scenes, Manchurian Candidate style. It's more intimate than the Imperius or mere Legilimency, and unknown and thus undetectable by conventional magic. And perhaps the unsuspecting student's thoughts and feelings have more sway over Voldemort's mind than he thinks...
Welcome to Hogwarts
Again, I'm sort of already doing my own version of this called Hogwarts Star Radio, but: see above. I've always wished there was more done with wizarding radio in fics, it doesn't get nearly enough attention.
An anonymous Hogwarts student starts a radio show for school news, detailing events in canon as they happen to Harry and company. Can optionally be an interactive fic, with commenters roleplaying as Hogwarts students or even other characters in the wizarding world.
Back Again
I've seen a lot of stories where Harry or Hermione or even Snape or Sirius go back in time to fix things in canon, but here are some unusual characters I'd like to see time travel:
Ron: I've hardly ever seen any fics where Ron's the one doing it. He doesn't get nearly enough appreciation, and I kind of want to see how people think he would handle that kind of crisis.
Fred (I imagine post-death, but that's optional): It'd be kind of cool to see him so out of sync with his twin.
Voldemort: Let's give the bad guy the worst possible advantage, shall we?
Peter Pettigrew (to before he betrayed the Potters): I want to see if you think he'd do anything differently, knowing how things turn out, and if he regrets betraying his friends or is purely selfish. And if he is purely selfish, how long could his relationship with the Marauders continue naturally before they realize it?
Fleur or Viktor Krum: I'd like to see what they would do to try and save Cedric.
Dumbledore (specifically back to when he first met Tom Riddle): Would he double down on his harsh scare tactics or take a gentler approach, like he did with Harry? How would he react to being young and healthy again?
Or alternatively:
Never Coming Back
A character from the past gets sent into the future! I've sort of seen this done, at least with fics where Harry's parents read the books, but getting to see a character with only pre-canon memories getting launched into canon (pun not originally intended, but I love it) would be really new and exciting.