It had to be Amy because, as a result of her growing up next to the crack in her wall, her mind became capable of resisting the cracks' effects with enough focus.
From "The Big Bang":
DOCTOR: Memories are more powerful than you think, and Amy Pond is not an ordinary girl. Grew up with a time crack in her wall. The universe pouring through her dreams every night. The Nestenes took a memory print of her and got a bit more than they bargained for, like you. Not just your face, but your heart and your soul.
DOCTOR: There's going to be a very big bang. Big Bang Two. Try and remember your family and they'll be there.
AMY: How can I remember them if they never existed?
DOCTOR: Because you're special. That crack in your wall, all that time, the universe pouring into your head. You brought Rory back. You
can bring them back, too. You just remember and they'll be there.
But perhaps more important is the fact that the Doctor specifically planted a trigger for Amy's remembrance of him in the form of the "bedtime" story that he told her before submitting himself to the crack in her wall. Knowing that she would one day get married to Rory, he exploited the wedding-day phrase, "something old, something new, something borrowed, something blue," as that also coincides with what Amy knows about his TARDIS. That trigger, combined with her cracks-resistant memory, allowed her to remember the Doctor back into existence just as she did her parents.
River did not grow up next to a crack, nor was she implanted with any mnemonic trigger, and so her mind alone would not have been sufficient.