If you just want to continue the function of the Miiverse it's in theory indeed possible to do so with custom servers.
There're some obstacles e.g. the communicaton between the WiiUs and the Miiverse is encrypted which makes it more difficult then e.g. with Wiimmfi. As far as I remeber Nintendo used a unencrypted connection back then, which even for that time was well lets say it was the Nintendo view on IT security
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Get some traffic from Miiverse actions such as posting or loading entries would be extremly helpful. It's also possible without that, but the amount of work needed to reconstruct the network protocol out of the binaries most probably would kill that project.
I don't see any obstacles which couldn't be overcome within the foreseeable future for creating a new Miiverse on newly written server software, but I don't see alot of hope for backuping the Miiverse from Nintendo in large scale. Nintendo offers a service to get backups from your own Miiverse content, but that hardly will be a large part of the Miiverse even if all people active in this forums contribute there old Miiverse entries to the new one.
Beside that I assume Nintendo just gives us pictures without alot of metadata. Metadata which is needed to place the Miiverse entries at the positions they belong in the new Miiverse. E.g. It's nice to get the drawing of your Yoshi for one of the Levels in New Super Mario Bros. U, but that dosn't means we know to which Level it has to be added. Maybe you're ready to create this metadata by yourself for your pictures, but I doub everyone is ready to do so... And some metadata is really hard to recreate e.g. if the comments are somewhere in the levels as it is with Mario Maker. Good luck to know the exact coordinates in such cases. For something like that we could write our own dumpers, but there're various problems with that.
Finally buying or renting servers would be needed (if enough passion for the Miiverse is there I guess there will be people to help out) and moderators would be needed too. I assume the first thing happening to the new Miiverse would be that alot of kids try to finally get there drawings of dicks into the games... Good luck to find enough people ready to do the moderator work.
All in all it's possible if some developer would be passionate enough about the Miiverse to make a new one, but... I wouldn't get my hopes up on that
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RedColoredStars:
