Haven/WMG

Audrey 2's amnesia was caused by exactly the same process that removed memories of Lucy
Not just the same force or agent behind it, rather, in both cases, all memories of Lucy were removed. The town doesn't remember anything surrounding the Colorado Kid, but a man who had mental illness and could understandably have been affected differently was the one who provided the name Lucy. The reporters know, but at least one of them appears to remember much more than he told Audrey. Then Audrey 2 has her entire life erased, but remembers basic things like how to speak, walk, eat, what a police station looks like, etc. She also remembers her boyfriend, who was specifically stated to have met her after the two Audrey's memories diverged. So every memory that would deal specifically with Lucy was taken, just in Audrey 2's case it was nearly everything.

Audrey is Lucy.
Because duh.
 * Pretty much proven, now.
 * But Audrey may still be Lucy's daughter. :D That line is a great example of why this show is good, despite mediocre special effects, and secondary characters who clearly aren't taking it seriously.
 * "Business As Usual" introduces us to the actual Lucy, revealing that history is repeating itself and many years ago, the situation with Lucy was similar to the situation with Audrey 2.

Audrey No1 is Lucy but...
Doesn't she have memories of her childhood? Maybe, at one point, she lost previous memories and gained Audrey's. Maybe she is her daughter with Accidental Fake Identity Syndrome. Maybe she's immortal. Maybe she's the origin of the Troubles. Maybe she's an alien.:) And we have to wait until summer.

Audrey No2 is from another dimension
With the introduction of another version of Audrey's boss, we can assume this is a possible explination.
 * As of S02E02 it seems much more likely that Audrey2 is actually the real one and Audrey just has her memories. Her Boss is not with the FBI but someone who tried to control her for unknown reasons. He is connected to the old Chief and knows more about the troubles and Lucy then he let on.

The Troubled are actually Alphas
What? It was obvious!
 * Related to this...

Haven is part of the Syfy-verse.
The town was built on some kind of Artifact (or it is an Artifact) that continues to give people abilities and make them Troubled. Some of the Troubled left Haven, had kids and passed shades of their abilities onto their children. Over the course of generations, superpowers became a large enough part of the gene pool that people noticed, and those people are Alphas. Naturally, this is all moving towards an Avengers-esque team-up.

History repeats itself and Lucy/Audrey ends the troubles each time
She is a power nullifier and every time the troubles start she ends them after learning how to control her power. While doing this she loses her memory and someone (perhaps the man who acted as her boss) gives her new ones. In each cycle she comes to Haven, learns about the troubles and her powers anew and stops them for a time. Before Audrey came the first time to Haven (or was born there) the troubles never stopped at all. The people with the tattoo are either trying to help/protect Audrey (possibly they are even the reason she seems to be immortal) or trying to stop her because they want to use the powers of the troubled for their own gain. This is the reason that Lucy seemed to use two very different hair styles - she had different memorys at the time.

Lucy/Audrey -starts- the troubles each time
The troubles start whenever Lucy/Audrey begins to remember or dig into her past. As she learns more, the troubles grow greater until she has to forget everything and start anew.

The writers read TV tropes!
In the audio commentary for "A Tale of Two Audreys", they talk about a few points made her: the darkness, the Velveteen Rabbit, even the averted unfortunate implication of Edi. Heck, they even talk about Fraudrey as Fraudrey.

Audrey/Lucy is Lucifer
Related to her being the source of the troubles, the whole point is a sort of gambit to try and get the Devil to be a good guy (or girl in this case).

Haven is linked to the Dark Tower series
It's Stephen King after all; a lot of his more recent stuff ties in directly, indirectly, or some how symbolically to The Dark Tower universe. And as of Season 2 Episode 11 we

Audrey's kidnapping is part of a plan to get Nathan to kill Duke or to be killed by him
Audrey is Tazered instead of shot, so she was taken alive. She falls there, but when Nathan arrives, he finds signs of a struggle over by the table, cupboards apparently ransacked, and Duke's necklace, which just screams frame-up. Since Nathan loves Audrey and doesn't trust Duke, he naturally assumes the worst and confronts Duke at gunpoint. Duke, as anyone would know, is on his boat where he has a number of guns hidden, and when the tattoo Nathan seemingly inherited from Max Hansen suddenly appears on his forearm, Duke has genuine cause to fear for his life. If Nathan kills Duke, he does what one half of Haven wants by removing the most recent Crocker threat to Audrey. If Duke kills Nathan, Audrey will never forgive him and he will be much more vulnerable to being influenced into killing Troubled people. And both sides know that Nathan loves Audrey and is quick to think the worst of Duke and that Duke gets nervous and somewhat trigger-happy when faced with a man with a tattooed forearm, but the anti-Troubled faction generally think Duke would win while the Troubled tend to have more faith in Nathan.

The Crockers CAUSE the Troubles.
That's why they have the ability to affect them on such a scale. And remember, this wave of the Troubles (as well as Lucy/Audrey's reappearance) only happened AFTER Duke came back to Haven.
 * Perhaps the Crockers are the power source of the Troubles so when they're killing Troubled, it's not that the troubles are going away, but getting absorbed into the Crocker family. Audrey - since she's anti-magic - in her various incarnations ends up getting close to a Crocker (proximity if nothing else)... which results in their ability getting negated and the energies releasing until they gather enough re-start the troubles. Audrey for her part may not be anti-magic so much as the reason for normalcy; just as the Crockers cause the troubles, she causes the non-troubled.