After reading through most of the comments about Wolverine being the one to travel back in time, I felt the need to post this do to the fact it makes sense in real life.
As we all know Kitty Pride was the one to travel back in time in the comic books, but having her do the same in the movie would be ridiculous. The time passed in the comics does not match up whatsoever to real life. In the comics Kitty Pride transferred her mind from her older self to her younger self. This would of course not work because she is a teenager in the original X-Men series and the movie for DOFP is set in the 1970s. While they could have made it her, it would only make things worse with the continuity.
With Hugh Jackman revealing Wolverine to be around 200 years old for The Wolverine, he makes the logical choice to go back in time using the mind transfer technique seeing as how Xavier might remember trying to recruit him in First Class. The only four characters alive from First Class to the original X-Men movie would be Xavier, Magneto, Mystique, and of course Wolverine.