JESUS CHRIST SO MANY BIOSHOCK INFINITE SPOILERS.
I am but a poor student who waits until games drop into the twenty to thirty dollar range before playing them, so please give me a spoiler warning before the next one of those, yeah?
Anyway, casually accidental time travel is all over the place in SotM - crazily, it appears to be by far the safer method. Consider: Chrono-Ranger and La Capitan just chanced into portals. Both of them eventually wound up with tech that let them control their movements in time, and are pictures of health (with the highest HP for non-powered humans among the heroes and villains, respectively). Meanwhile, Omnitron-X actually designed its current body specifically for time travel, and still wound up being a superpowered robot more delicate than a human being. And Visionary... she covered the shortest temporal "distance" of all of them, and for her trouble, got a deadly brain condition and an evil alter ego.
Miss Information's circumstances are, obviously, a bit unique, but I see no reason to assume she's not just another random chance time traveller (presumably, sent back by the event that "killed" her) who has learned to use this to her benefit... as far as canon is concerned.
For the ongoing storylines of your own game, though, Miss Information is a bit of a quandry, in that her deck as designed really only makes sense once. "Oh, some's sabotaging our stuff again - do you think it's Aminia? I thought she had forgiven us after the third time we re-hired her after she almost killed us all." So, if you dig the character, having her start to be La Capitan is a good way to keep her around.
For my own part, I've taken the opposite tack. I play Miss Information's starting side, defining it more nebulously as "the traitor". Then, instead of flipping when the time comes, I switch to a different villain - the team doesn't know someone was messing them up as an inside job, they assume everything was somehow done by Blade or the Ennead or whomever. And I am working on some variant rules so that when the Traitor is revealed, it's one of the Freedom Five, themselves...
Ironic