In the movie Gone Girl, Amy has killed a man. She has his blood all over him. The cameras in the house would prove that Neil’s character was innocent. Even the smallest of investigations would prove her guilty or at least cast suspicion on her. What made the FBI trust her when even the detective doesn't trust her?
What the film maker's want us to believe:
She was clever.
You say the cameras in (correction: outside) the house prove Neil's innocence. How? They show a man turning up to a woman who is in permanent confinement in a house, bringing her clothes and food, but never leaving with her.. His actions certainly don't look entirely innocent - it suggests she is a prisoner inside.
Couple that with her staged escape attempts, where she pretended she was a prisoner/tied up for the cameras, and suddenly he looks like a dangerous suspect who has imprisoned this poor woman.
She plays heavily on this and, in his death scene, acts as if she has been raped and has had to kill him to finally escape her captor.
Of course, you could argue that for a long time she wasn't in the house (when she was missing), but this is largely irrelevant - he could have been keeping her anywhere during this time.
A perfectly plausible diverging opinion:
It's certainly arguable as a plot hole.
She arrived at the house willingly and there would be footage of him every day leaving the house for work and her doing... nothing. Not escaping, not running, not fighting. Other than her one apparent staged attempt, she's never near the doors - and likely there would be some footage somewhere showing her just inside the threshold of the property, totally calm and nonplussed.
The NY Post certainly felt this was a plot hole. Another arguable issue was that if he was shown to be in the house during her supposed abduction, he would have a solid alibi. Alas - none of this appeared to matter to the film-makers and it is never covered.
The cameras are all outside the house.
The reason she performs the bits of theater that she does (spilling the wine, pretending to tie herself up), is so that she will appear on the external camera as someone who is being held against their will.
There's no footage of the actual murder taking place, but there was evidence that she had sex with her "captor" (which is why she waited for him to finish before killing him), so all video footage will backup her claim of kidnapping and rape.
The only possible exception is her initial arrival, when she was in disguise, and not yet pretending to be held captive, but she could easily say that she was terrified for her life at that point, and trying not to provoke him.
There is another possible explanation, though I also think this was a big plot hole. Home security video surveillance systems don't keep recordings forever. At some point, the videotape (on older systems) or hard drive (on modern ones) gets full of video and older footage needs to be overwritten by newer footage. It's conceivable that Amy's arrival at the house was captured on video that was later overwritten by newer video, and that the only relevant footage that the FBI found was her later staged performance.
However, it's obvious from the movie that NPH spared no expense when it came to his house and I really doubt that his video surveillance system didn't have enough capacity to record for months on end without recycling. So though this explanation occurred to me, in the context of the movie I don't buy it.
I also seem to recall Amy saying during questioning at the end of the movie that she had been abducted by NPH much earlier than she actually arrived there, like from the beginning of her disappearance. The surveillance video from the time before she actually showed up at NPH's house would presumably have proven that wrong, showing NPH going about his life as normal when he was supposedly abducting Amy and keeping her prisoner. That was ignored too (if I remember the ending correctly).