How about the following as a conception of evidence, based on Bayesianism? X is evidence for Y if X is more likely to exist given Y than given not-Y.
Now if we accept this, it looks like there are lots of things that qualify on behalf of theism and on behalf of atheis. Beginning of the universe? Maybe it can be reconciled with atheism, but it's not what an atheist would expect. Ditto for the fine tuning of the universe?
Evil and suffering? Sure it's possible given theism, but is it more likely given atheism?
With this model, we might say of our opponents that there isn't enough evidence, or that the evidence is outweighed by the other side, but can we really make the "no evidence" charge?
Is there something wrong with this definition of evidence?