The idea of the story is that there are no evidence.

But in the OP, you said:

she has known Adam very well and the Adam that she knew, wouldn't commit murder. She uses Adam's character and her personal relationship with him as evidence.

It seems to me that Adam's character as observed by Eve is evidence. Not irrefutable evidence, but evidence all the same. It seems to me that, baring evidence of Adam's guilt or evidence that Adam's character had recently changed, Eve is rational for beleiving Adam to be innocent on the basis of that evidence.

Cain provided no such evidence, so Eve is rational in her belief.