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Not sure I'd even bother with a backstory. Just start the story fresh, not waste a second of film time explaining 'what happened'.

Maybe have Ripley, Newt and Hicks suffer from strange headaches and let the audience fill in their own backstory.

Sgt Jenkins: "You feeling okay Hicks?"

Hicks: "Yeah, just a little dizzy....Lets kick some Xeno ass!!!" *Blam! Blam!*

Now that it seems appropriate, I'd like to end my discussion with the following quote:

Hold fast to dreams.

For if they die, life is a broken-winged bird.

That cannot fly. 

I, personally, think NB should go: "it was all a Conspiracy" - which makes considerably more room for creative headway than: "it was all a Dream," - you can, however, still have the "dream-theme" that has been consistent throughout the Alien franchise, and have the 'conspiracy theory' (we'll call it "CT" for short) plot simultaneously. It would go something like this:

Alien Resurrection and Alien 3 no longer exist... except within Ellen Ripley's own mind. 

Outside, in the 'other' reality, the one Alien 5, Aliens and Alien take place in - certain events like the death of Newt/Hicks, are symbolic of what really happened to them in the CT Alien 3, and the clones/hybrids in Alien Resurrection are symbolic of and correspond directly to what happens to Ripley in Alien 5. 
Then connect the dots:
1) Hicks/Newt - She finds out they are alive and well, and think she's dead (like she thought about them, except it was Ripley that "died" - I'll get to that in a moment, bear with me).
2) The Dreams - She finds out corporate elites/the military have been experimenting on her memories, studying her 57-year hypersleep, experiments studying immortality. The company, we'll call it "Zhang-Dykstra (ZD)" is run by a secret society of Alien cultists that worship the Hive Consciousness as god, and that ultimately they are the final step in evolution and that all will be unified within a Lovecraftian Hive Dimension Superorganism, of sorts.
3) The Story - Hicks and Newt are the male and female leads respectively. Only we don't know she's Newt until towards the end. We're led to believe she's the daughter of billionaire and ZD Rep. Charles Rutherford - a man who crossed Mike Weyland (Bishop 2) and connects her even further with Ripley and the experiments the Company is running on her. 
4) Horror - It has to be utterly terrifying. I mean the DARK kind. Think Poltergeist in space, as opposed to Texas Chainsaw Massacre in space in the essence that Poltergeist is more cerebral about its horror than TCM, the latter of which is a slasher. Like hands and silhouette, bizarre movements, disorienting claustrophobia, frighten the audience with immense strangeness (I can't stress that enough) as opposed to gore and cheap jump-scares that audiences are largely desensitized to now-days. 

And most of all, a big MENACING Alien. Not the little pipsqueaks we got in the AVP films. I mean an adversary that gives the impression of a unstoppable force of nature, whose very nature itself only becomes more mysterious the more we find out about it and the nature of the so-called "Hive Consciousness". Maybe tie in the power-vacuum (no pun intended) between the Alien (1979) and the Aliens (1986) that we notice between the two films.

What are they studying Ripley's dreams/ memories for?

@S.M

There is an infinite number of answers to such, one is perhaps to create emergency protocols to fit the reactions of a victim of a xenomorph. Or to study subconsciousness or instinct.

Are any of these infinite numbers of answers particularly viable?

I'm willing to be surprised if Blomkamp went down the dream route and it wasn't utterly creatively void, but I've yet to hear any 'those films were a dream' scenario that wasn't precisely that.

Having them as being dreams would feel sort of cheap I think. Maybe AR could be used as an alternative timeline after A3 like: this could have happened but it didn't (but that is because I like A3).

'I'm willing to be surprised if Blomkamp went down the dream route and it wasn't utterly creatively void, but I've yet to hear any 'those films were a dream' scenario that wasn't precisely that.'

Thats the problem, 'it was all a dream' was a cliche decades ago. I'm dubious about A5 already, if it went that route it would just be embarrassing.

I got a kick out of Newt saying 'Can we dream?'. Ripley should have said 'With luck, we won't dream at all.'

The only way they can do it I believe (other than setting it after Resurrection), is to just flat out ignore Alien3 and Resurrection.  Never mention them, never allude to them.  Those films were a possible future.  Blomkamp's film is different possible future.

It would be interesting if there was a multiple dimension angle to proceedings and the next incarnation we get is an alternate timeline/parallel universe to 'our own' Alien universe that cause and effects go both ways so in alternate universe (Bloomkamp Universe) the Ripley there is "ghosted" space/time-temporal displacement-wise with Alien 3/Alien: Resurrection Ripley and vice-versa. Each one apparanly having visions of the others temporal experience in their own respected universe.

So for example, Ripley wakes up having a nightmare in Aliens as a result of her experiences in the event of Alien, it would be interesting in the third incarnation of the series in-keeping with the "dream-theme" there is a scene-or-two where Ripley does indeed have nightmares that are ghosting effects from her temporal counterpart in the events of Alien 3/perhaps Alien: Resurrection.

Man, do I love this post!!!