Time can be rewritten, that's a running theme this season. And we know he gave her some sort of information when she was younger that he needs her to remember. So he might have given her something personal enough (example his name) that if she knows it he knows she's on the up and up.
Assuming that River whispered his name (to 10, in the library), and if the Doctor tells Amy his name, then my Water=Water theory may be right, Amy POND is RIVER Song in the future, left in the 52nd century after their journeys for some finagled reason. I doubt the Doctor is just gonna start giving away his name to everyone and their mother now.