While it wouldn't be unreasonable to want to see some of the classic villains or misunderstood adversaries of TOS, I kind of want to see something different. Maybe a race of people that were not discovered in the "prime" TOS timeline but now have to be dealt with. The problem with using some of the characters or races suggested is that we all know that Kirk and his crew faced them and defeated them so it really wouldn't generate any suspense to reuse them. All you would be doing is updating a TOS episode and maybe changing the method that the crew uses to defeat them.
At the same time, I think that in this nuTrek timeline (whatever you want to call it) it's very likely that this crew will have some of the same adventures and face some of the same villains because let's face it how many lives were really altered as a result of Nero's actions? Obviously Kirk's life was affected but he somehow managed to achieve his destiny despite the altering of the timeline. I'm sure that if any of these villains lives were at all altered the same might be said for them too.
Ultimately it all comes down to a creative decision. You can either establish that the crew faced some (maybe not all) of the same adventures that the "prime" TOS crew faced and update the TOS episode or episodes that the next film will be based around (which really wouldn't be that interesting if you ask me) or you can eschew all of the established "prime" TOS plots and villains and give this crew a whole different path than the one that TOS followed.
One way to achieve a middle ground is to perhaps show a montage of this new crew handling some of the same conflicts that the TOS crew did and establish the similarities between the two universes before going into a conflict that never occurred in TOS. So in theory what could happen is we see quick glimpses of Kirk and crew facing some of the same villains but the main story would focus on something no fan would expect. I think maybe that's the best way to go.