For me, the greatest mystery in physics is dark energy. It is inexplicable within today鈥檚 physics, though we expect it to be related to the origin of our universe and the nature of quantum gravity. Will we discover its identity by 2056? Maybe, though we will need a brilliant new theoretical idea.
It could be that dark energy reflects the anthropic nature of our universe, which implies there are other universes. If we could get evidence for their existence, this would be a remarkable breakthrough.