I'm sure he recognizes the odd behavior from Amy, but how does he remember the Silence without an eye drive?
There is nothing that prevents a person from knowing of the Silents' existence.
While you cannot remember anything that you do in the presence of a Silent, that does not prevent you from, say, reading about them, or hearing about them. That was the whole point of the nano-recorders and the tally marks in "Day of the Moon"; they allowed the Doctor and his allies to circumvent the Silents' memory-proofing. A recording or a note that you leave for yourself while looking at a Silent (one of which Amy had photographed with her phone way back then) isn't going to go away just because you stop looking at the Silent, so you can then read said note and ascertain the information independently of a Silent. So it is entirely possible that the Doctor simply wrote down everything that he needed to know while looking at the photograph on Amy's phone (the first scene on the TARDIS in "Day of the Moon" would have us believe that knew of the picture on her phone, since he had by that point set the TARDIS up to extrapolate the image and create a hologram of it), looked away from the phone, and read the note that he left himself.
Furthermore, in "The Time of the Doctor", the Doctor was having regular meetings with Tasha Lem, who was the head of the Papal Mainframe. As the Silents were the priests of the Papal Mainframe, it seems unlikely that the subject would never have come up in conversation between the two.