As arguably DC's biggest character, there's not a whole lot we should change about Batman. I mean, he's Batman. The guy's awesome just the way he is.
Now, the world around him does need to change a fair bit. Starting with one of his most famous enemies.
Joker must either die or be massively reinvented.
Don't get me wrong, I love the guy as much as anyone. However, the character has been pushed too far. We've had thirty years worth of writers trying to have Joker do something even more horrifyingly evil and depraved than Death in the Family and The Killing Joke, and it's just gone too far. There's a reasons fans have argued for decades that The Killing Joke ends with Batman killing Joker. After all the horrifying things Joker does in that story, the idea that he just gets tossed into a cardboard prison only to escape and do it all again a week later just isn't a satisfying ending. Now to be fair to Moore, leaving Joker alive fits with the story's tone about just how insane the relationship between Bats and Joker is.
Problem is, that story became the gold standard everyone wanted to copy for him. We get stories like Joker's Last Laugh and Emperor Joker where he kills millions or even billions of people, and at the end he just gets tossed back into the same easily escapeable cardboard prison to do it all again next week.
Thus, he either finally dies (and stays dead for a significant time) or gets rebooted. If the latter, I'd change take inspiration from the DCAU. Keep him on theme and dangerous, but without the body count or ridiculously gruesome antics like cutting off his own face, then stapling his decaying skin onto his face (Seriously, WTF DC?).
However, I'd prefer a dead Joker, just for all the story options it opens up. That would be a huge way to hook in audiences for the reboot: End issue 1 with the revelation that Joker has been killed. We get a classic Batman detective story as the dark knight tries to find out whodunnit, with the extra twist that half the suspect pool comes from his own circle. Just about every member of the Bat Family has very good reasons to want Joker dead, as well as just about everyone else in Gotham. Hell, have him carry on the investigation in the face of other characters saying it doesn't even matter who killed him, or that if they found out they'd thank the killer.
---
And now, on to the Bat Family. Which will be getting a bit broken up for its own good.
Dick Grayson is Nightwing. He has dealt with his many Bat-Dad issues by becoming a Teen Titans character who doesn't do Batman stuff.
Barbara Gordon is Oracle. Keep her in the chair or have her leave it, but she's far more interesting leading the Birds of Prey than as Batgirl. As for the Birds, I'd keep the classic lineup of Oracle, Black Canary, Huntress, and Zinda, with the rest of the supporting cast moving in and out as suits the story. More Savant and Creote can only be a good thing.
Jason Todd can occupy the grey area he does in stories like "Under the Red Hood." A vigilante like Batman, but showing the darker side of that. Most RL vigilantes were not nice people. He can make a great contrast to Bats, and constantly skirt that line without ever leaping over it.
Tim Drake stays on as Robin. He was a good Robin, and someone needs to fill the role.
Cassandra Cain was the best Batgirl. It is known. Why ruin a good thing?
Stephanie Brown sticks with being Spoiler. Spoiler is awesome. She can bounce between the Bats and the Birds of Prey—maybe even make her a sidekick to one of the Birds. If nothing else, seeing her and Misfit bounce off each other should be grand.
Damian Wayne should be Nightwing's sidekick. Their chemistry is far better than Damian and his father. Easy enough to justify too—Bruce is self-aware enough of his failings to know his son needs balance in his life.
The Rogues Gallery is mostly fine. Just use the reboot the remove the couple "WTF were they thinking?" stories they build up over the years.
The biggest change I would make it to continue the trend of nudging Harley and Ivy together, and in a bit more of a grey direction. They make a very good double act, and seeing them bounce between anti-heroing and villainy would be fun. Plus they're probably one of the most well-known homosexual couples in comics (Even if DC only officially outed them in 2015, the writers were making it clear from their first team-up in BTAS), and I ship them. Two very compelling reasons to keep them around.