I think that to give a well-balanced answer, we need to consider what Conway intended the phrase to mean. Now, without further statements from her, we can't really know what was going on in her head, but I think we can give her the benefit of the doubt and rule out:
- She was naively putting up a weak excuse for "lying".
- She was calling it a false statement that (for whatever reason) the administration wanted, or wanted the public, to assume was true.
So that, as far as I can see, rules out most existing answers (certainly the highest-voted ones). I'm not saying those answers are wrong; I just think that other answers are possible if we optimistically assume good faith from Conway.
What are those answers? Well, the one that occurs to me is this: she may have meant that...
Both sides were starting from different facts and drawing very different conclusions.
In this interpretation, Conway was conceding that the media had started from factual information, but she believed that their information was leading them to incorrect conclusions. She meant that Spicer's facts were equally correct, but that his conclusion was better.
The classic parable of the blind men and the elephant, which LamonteCristo's answer contains an illustration of, is about exactly this phenomenon. All starting from equally correct facts, all coming to plausible conclusions that can't all be correct (though in the parable, they're all wrong).
I realise that this is largely speculation on my part. However, I believe this still constitutes a satisfactory answer to the question: "What could 'alternative facts' mean in English?" It can mean facts not yet considered, or not given much weight, which change the conclusions people are likely to come to.
Edit: I would appreciate feedback from those who are downvoting this answer. I'm not trying to take Conway's side by giving this answer; I'm talking language, not politics. Whether she was telling the truth or not is not directly related to the meaning her words had.
To put it another way: If I say "the sky is red", I mean "the light reaching our eyes from the atmosphere overhead causes a response in the eye similar to that of light of approx. 700nm wavelength". Knowing that I'm wrong doesn't change the denotation of what I said. And whether Spicer's "facts" were true or not doesn't make my suggestion, that Conway was claiming they were factual, any less plausible.
Edit 2: Of course, now that Spicer is on the record as saying "sometimes we can disagree with the facts", my inference is even less likely to be what was meant. I stand by it as an answer regarding English usage, but in context it doesn't match with the evidence.