Obviously I have no evidence to back the following suggestions up, but neither does other more or less accepted hypothesis, like the 'inside black hole' one, or the 'multiverse' of M-theory.
Suggestion 1:
Could a new big bang have occurred 5 billion years ago, inside our universe, causing our universal expansion to accelerate? Like a balloon inside a balloon?
Suggestion 2:
This one should be easier to dismiss.
Could the big bang have happened in two stages? Stage one releasing only dark matter, and later the version we know, with both matter and dark matter? In that way our universe would be surrounded by a huge gravitational field, that is only getting stronger as we are catching up to it?
I myself lean more towards the 'biased measurements' solution, but until we're sure, I take the acceleration very serious.
I apologize if you think this is a complete waste of your time.