From Memory Alpha:
In 2258 of the alternate reality, Section 31 performed secret investigations in unexplored space for means of better defending the Federation after the destruction of Vulcan. They discovered the Botany Bay, 262 years after launch, and Khan was awoken. With the other 72 augments still in stasis under Section 31's control, he was forced to join and work for Starfleet as John Harrison.
Admiral Marcus: I took a tactical risk and I woke that bastard up, believing that his superior intelligence could help us protect ourselves from whatever came at us next. But I made a mistake. And now the blood of everybody he's killed is on my hands.
As I said; it would've been far better off to leave out the whole Khan-connection, and have John Harrison BE John Harrison. Just another augment. Only, like other augments (such as Khan), hell-bent on controlling the "lesser" humans of Earth. In both realities (TOSverse, and JJverse), the Botany Bay is discovered (Section 31/Enterprise respectively). If a different person/group finds it in the two separate realities, why would they choose to unfreeze the exact same person? Reviving Khan (in the TOSverse) was an accident (they had no knowledge of whom they were reviving before they revived him). It appears to me, that Section 31 chose to specifically revive Khan.
And by having "Harrison = Khan", they made it all the more confusing, by having a white British actor playing Khan, compared to Montalban's version who was considered a Sikh (from India). Ofcourse, the JJ comics explained Harrison's appearance as having been genetically modified to further blur his true origin from others.
I like the JJ movies (as Entertainment). But as Trek movies, they still leave much to be desired. Not the least of which being, the fact that many crucial plot details are left out of the movies, and instead added in a secondary source (comics). Harrison's origin explained in comics for example. Nero's origin and reason for hating Spock was in the comics as well.
I firmly believe that Trek's future is in television. Where writers can take their time to properly flesh out characters, tell detailed stories etc. Movies are focused on Action, Explosions, and being fast-paced. Most Trek movies (not just JJ's movies) are centered on action, and less on character development.