Rational evidence is the broadest possible sense of evidence, the Bayesian sense; rational evidence about a hypothesis H is any observation which has a different likelihood if some hypothesis H is true or alternatively false.
Rational evidence is distinguished from narrower forms of evidence, such as scientific evidence or legal evidence. For evidence to be scientific, it must be, in principle, publicly accessible; you should be able to do an experiment to verify the knowledge. For evidence to be admissible in court, it must meet other legal standards.
For example, suppose I tell you that the original author of this paragraph wore white socks while writing it. (In fact, I do so tell you.) You now have rational evidence that the author of this paragraph wore white socks. But it is not scientific evidence because there is no experiment you can do for yourself to see whether it is true. And it is not legal evidence - you could testify in court that I had told you my socks were white, but you could not testify that my socks were white unless you had observed it for yourself.
Rational evidence in the broadest sense may be useful only for private thinking, or thinking in the company of trusted friends. Scientific, political, or legal processes may (for good reasons) operate with less trust, and impose special and additional standards of admissible evidence. The scientific method can be viewed as a special standard of admissible evidence protecting a pool of extra-strong beliefs.
Conversely, a belief can be rationally knowable without generating the sort of specially strong evidence that would qualify it as "scientific", just as a police detective may rationally know the identity of the city's criminal boss without having the special evidence needed to prove it in court.