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 depending on whether H holds in reality or not.
Rational evidence is distinguished from narrower forms of evidence, such as scientific evidence or legal evidence. For a belief to be scientific, you should be able to do repeatable experiments to verify the belief. For evidence to be admissible in court, it must e.g. be a personal observation rather than hearsay.
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.
The scientific method can be viewed as a special standard of admissible evidence protecting a pool of extra-strong beliefs. Conversely, a fact can be rationally guessable without it having generated the specially strong evidence that would qualify knowledge of the fact as "scientific"; just as a police detective may rationally know the identity of the local crime boss without having the special evidence needed to prove it in court.