Here we go, a new fan-fiction using a setting I haven't seen anywhere on the site. I'm happy to present Deadman Wonderland: Rebirth as the first of its kind. Although the joy is hampered by Webnovel's/Qidan's dumb decision of changing the way fan-fiction works, quarantining it into this strange category of "others" which you can't really access. Anyway, let's just focus on making this idea good.
So first off, instead of talking about possible powers for the protagonist we're going straight into who the MC could be. There is the obvious choice of reincarnating as Ganta, which would cause a heavy amount of the plot to change, though that's not necessarily a bad thing. It would hasten the development of Shiro, causing the fusion of her and Wretched Egg's personalities to happen much earlier, or even preventing the split from happening in the first place. While that's a splendid way to start, I actually would like to see a new character be born instead. There's a few possible choices to that end, and I'll talk a little about each one.
Number one is either an older or twin brother to Ganta, firmly establishing a familial bond between the two, and guaranteeing a friendly relationship with Shiro, possibly later evolving into a couple. All of these scenarios will need to have the MC improving his body as blood is a required medium for the Branches of Sin.
Two is being childhood friends with both Shiro and Ganta, which is a more reasonable choice for ease of writing. You can introduce the protagonist right before the experiments on Shiro, who will use both as moral support, preventing her mental breakdown and stop her from attempting suicide. It's completely up to you if you want to use Shiro as a possible lover, or try and find or create another. While this series has a heavy focus on the traumatic, it does still have a few tender moments in-between.
Finally, we introduce the MC as the brother of Shiro, which eliminates her as a candidate for wife/girlfriend. Shiro was adopted by Ganta's mother as a way to avoid using him as a test subject. With the protagonist present, he instead becomes the target for testing, while Shiro stays a close friend of Ganta, removing any possibility of her losing out on a normal childhood. She remains mentally stable, and enters the same middle school as him. If you choose this path, the MC will basically have the same abilities as Shiro in the original story.
So, what kind of Branch of Sin do we give the protagonist? Well, I thought about it, and really the only thing I could think of that would be balanced but still cool enough to be an MC's power is to basically give him the ability to mimic other Branches while also coming up with his own variants. Of course you could also give him increased regeneration, which will help with the anemia that Branches can cause.