Ben'smagic box - whatever one wanted to be in the box would be there when it's opened. John secretly wanted Cooper to be in the box so he could get revenge on him. In reality, Cooper is in the outside world alive and well. He's up to his old tricks and thinks his son John is dead.
Ben's magic box *is* the Monster. It can manifest people who are still alive (Walt) animals or things it has never seen but which are important to the characters (Kate's Horse) and can be summoned by Ben. It took the form of Anthony Cooper, who is still alive in the real world.
Very likely now that we know more about the nature of Jacob's nemesis.
Not an entirely satisfying explanation. In no other place does the Monster appear to die.
Cooper's spirit, assuming he actually died as a result of his car accident.
A manifestation that was manufactured in the "magic box" of Jacob's cabin.
Coopers arrival on the island will be explained in a Richard Alpert flashback.
It will be explained in a minor Others flashback.
It will be explained in an unknown Others flashback.
One of the Oceanic Six will found out what how Cooper came to the Island.
John Locke in the real time line needed to kill his father. So the future John Locke went back in time, found his father and sent him to the Island.
Cooper is Jacob's nemesis in disguise. He needed to impede Locke's ascent to leadership. Had Locke become the leader too quickly, he would have asked to see Jacob. Assuming that Christian Shephard and the man in Jacob's cabin are related to Jacob's nemesis, Locke and the Others would have had known not to trust them and Nemesis's plans to kill Jacob would have been thwarted.
Smoke Monster Manifestation
It seems very likely that he is a manifestation of the Monster (various forms) given first the strangulation at the Black Rock and the current state of affairs i.e. Flocke and Sawyer tag team.
Cooper mentions his other aliases (Adam Seward, Ted MacLaren, Louis Jackson and Paul-something) in "The Brig". These names will be relevant to another survivor in a future episode.
Possible cultural references
The last name of Cooper is a reference to D.B. Cooper. He's the right age, a meticulous planner and certainly ruthless enough.
Interestingly enough, in The Brig Cooper says that he didn't raise "fools" when referring to Locke, rather than "fool" in the singular, possibly implying that he knows of his paternity of other children. Similarly, he seems to (genuinely) question whether or not Sawyer is his biological son, further indicating that he's aware of fathering multiple children. From "The Brig";
Anthony: Guess I didn't raise no dummies
One of his children are also on the island (like Jack and Claire).
Kate is the daughter of Anthony Cooper.
Anthony was married and had other children he didn't raise besides Locke. From "Deus Ex Machina":
Anthony: You have a family your own?
Locke: No, sir.
Anthony: Me either. I tried it a couple of times, didn't take.
The name of the Jean Shephard (a real-life person, no relation to Jack or Christian) song is "Mama Didn't Raise No Fools." It's a figure of speech.
Anthony Cooper is just an alias. In "The Brig", he lists some of the names he's gone by and "Anthony Cooper" was right in the middle of his aliases, thus hinting that it's not his real name.
His real first name is Roger. This is suggested as John Locke's mother's name is Emily, the same as Ben Linus, and connections have been displayed between them. ie, their connection to the Island, their passion to stay on the island, their thirst for purpose/leadership.
There do seem to be a great deal of similarities between Locke and Ben. Perhaps that's the only reason Ben was allowed the shogun-like authority he as is because he was close enough Richard thought this was the one they were looking for after Locke seemingly failed the original "Lama test". Though it is possible Locke didn't fail, he just wasn't ready to go to the Island as The Purge had not happened yet so perhaps the Island wasn't ready for him.
Anthony was rendered comatose in the car wreck that brought him to the Island. The Island healed him, allowing him to regain consciousness, but the Island's destruction in the other world prevented him from ever regaining consciousness after the incident that caused the coma in the Flash-Sideways universe. Anthony being put in a coma at all (and thus being unable to work toward moving on until he wakes) is a result of his unwillingness to accept the existence of Purgatory (cross-reference his insistence that the Island was Hell), but the reason he never wakes up is because the only thing that can wake him is a disruption of nature, spirit, and soul like the (non-existent) Island.
He is not Locke's real father, he just used that as a ploy to get the kidney, and then convinced Emily Locke to lie about him.
You can't just get a kidney from anyone but need a close genetic match. Cooper could have secretly done tests to verify that Locke could donate a kidney to him but it is more likely that he is actually Locke's father.
He is the one who Locke hears saying "Help me" in Jacob's cabin. Jacob is Ben's personal projection just like his dead mother was when he was a young boy and saw her physically appear on the Island (not knowing then that he himself brought her there by aching for her to come and rescue him). For Locke to hear someone who has physically materialized from another man's memories doesn't make sense. Ben can't hear the message Locke receives from his father any more than Locke can hear the conversation between Ben and Jacob. Seeing as Locke arranged to have his own father killed, it would be logic if it was the desperate call for help of his father that he heard.
The producers have already stated that Cooper was brought to the Island by the Others so this doesn't hold water, nor do any other theories about the "magic box". It seems to have further been stated that the "magic box" is a metaphor as opposed to an actual box. This would further negate theories that Jacob's Cabin is the box. Any manifestations seem to simply be the product of "The Monster" as it has been shown scanning people's thoughts and memories previously.
The magic box is just a metaphor as explained by Ben.
Locke's private investigator was compromised and Anthony led Locke to believe he was his father so he could take his kidney.
When Cooper was 'bonding' with John, this was not just for the sake of the con. Locke's real force on the Island has been as a hunter; the time he spent with his father was a sort of training or engineered preparation for what was to come on the Island.
Further to the "hunt bonding", Cooper stealing Locke's kidney aided in Locke's healing after he was shot by Ben. "If I still had a kidney, I would have been dead" Locke S4E2. Perhaps Cooper engineered the missing kidney because he knew Ben will shoot him in the future.
He was brought to the island by the Others as a means for Ben to manipulate Locke and/or Sawyer.