So like most people, I guess it's the father that was the second person (since he's shown remembering bits and pieces while dreaming, possible lingering effects of the magic). It's also reasonable why he would choose for her to remember instead of remembering himself.
The key point to remember is that the world would go to ruin without the saintess. Also, the consequences have to be grave enough for the magician to agree to turn back time, since he personally can't take advantage of doing it without relying on a second person to fill him in on the details of what happened. I personally believe Cosette is an agent of the demons sent specifically to eliminate the saintess, with others working from the shadows to lower public opinion of Keira and demand her execution despite her not having commited any crimes. So after she is executed, the kingdom falls to ruin. This gives the sorcerer the motive to turn back time. Still, he needs someone to send back who has the power to prevent what happened.
Now, it should be obvious to everyone that Keira was the real saintess after she was executed. But how can someone sent back in time convince everyone that Cosette is the fake? Remember that Cosette was able to outmaneuver Keira, and also ingratiate herself with high society and had the support of whoever wanted Keira dead. Some random person accusing her would have been swiftly dealt with. The obvious person to send back is the one with the most authority, her father. If he remembers the future, he can simply change how he acts, not execute Keira, imprison Cosette, find the people who supported her, and unravel the whole conspiracy. He knows everything that happened after the execution, so he would know exactly where to strike, and as the duke he has the means to do it. If anyone should remember the future, it is him.
But, he is a coward and an idiot. He probably couldn't stand the guilt of having executed his own daughter, the one who struggled for his affection for 18 years, after falling for the lies of the fake he knew for about three months. The thing he wanted most was to forget his own failures. So, like the coward he is, he shifted the responsibility on to his daughter. He basically risked the future repeating itself, just to spare himself the grief. And he handed over the grief of being executed, and the responsibility of avoiding that future again, over to his daughter. Who wouldn't know what happened after her execution, wouldn't know about the pact with the sorcerer, and would generally just be extremely confused, hurt, and terrified.
Anyone else would have chosen for the duke to remember, because that solved the situation immediately. Making Keira remember just means that the victim of the conspiracy, who fell for it the first time around, must now unravel the conspiracy while being targeted by it. The only person who would choose to have Keira remember is her father, since he is the only person who would want to forget the future.
