What possible evidence can there be for gods?

I see that some rationalist non-believers (such as Jerry Coyne) say that unlike some other rationalist non-believers (such as PZ Myers), they believe that there can be evidence for the existence of gods.  I'm really trying to understand this position, because it seems to me that the supposed nature of gods necessarily includes attributes that are beyond any evidence, ensuring that there is a vast difference between an enormously powerful being who can manipulate the world and a god.

Some of the attributes that may be assigned to gods are:

1. Being an avatar in some way of Nature - Poseidon isn't just good at making storms, he the personification of the sea.   Poseidon and seas are somehow bound together.

2. Being the source of what otherwise would be an abstraction - Yahweh is the source of morality, not just a good being.  Some say Yahweh is the reason for mathematics working.

3. Having creative power.  A god may not be a being within Nature, but may instead be the source of Nature, in some sort of 'ground of being' way.

4. Having magic abilities - being able to work miracles.

I'm at a loss to understand how any of these attributes can be supported by evidence.  A powerful being might be able to do very dramatic things, and might say that they are the source of morality and logic, but there is no way I can see that such claims can in any way be demonstrated by evidence.

This matters, because the difference between a hugely powerful being and a god is that in religion a god gets to define what is right and wrong, and people live their lives accordingly.

I can see how there might be evidence that a powerful being performed various acts that were interpreted as miracles by the authors of holy books.  But that still doesn't get us to that being having the kind of moral authority that should influence lives.  Godhood matters, and yet it seems to be that godhood is clearly beyond any possibility of demonstration by evidence.