On PHB p. 122, it says:
"If you have the Combat Reflexes feat ... you can add your Dexterity modifier to the number of attacks of opportunity you can make between actions."
This strongly implies that resetting on the turn boundary is hogwash.
In KD's example, I'd rule the AoOs recharge on round 2 initiative 12. The Ready" on 1/25 is the end of your first turn -- your next turn starts on 2/12. PHB p. 134: "The ready action lets you prepare to take an action later, after your turn is over but before your next one has begun."
So it's clear an extra personal turn/round isn't being generated by the ready.
I think it's pretty clear that AoOs are meant to reset every personal round. It really only makes sense that it happens at the beginning, because otherwise one may ask "when does flat-footedness go away?"
That is, A starts his turn flatfooted. He moves past B drawing an AoO. B decides to do an unarmed attack on A, who is armed. It seems obvious that A can indeed take the AoO, that they charge up at the beginning of A's action.
All of this is a bit overkill.
Of course AoOs recharge at the beginning of each of your turns

Interpreting "round" a la 2E is kooky, "round" a la spell duration is a hassle.