Cherry-picking refers to selective presentation of evidence in an argument in order to refute or affirm a point while ignoring other evidence which will not support the point(s) being made. It derives from the obvious reluctance to harvest unripe, or overripe, fruit and to select only those which you will consume.
Often, cherry-picked factoids or references will be over-extrapolated and oversold to give the impression that they are representative when they are not.
Cherry-picking often relies upon anecdotal evidence, because it only uses one or two examples to make a point; statistical cherry-picking essentially use larger-scale anecdotes, by ignoring the broader evidence on an issue.
Quote-mining is a form of cherry-picking, and the genuine points used in the construction of straw man arguments are typically cherry-picked.
Dr. Jones and Dr. Smith published an article in the Klingon Journal of Vaccine Evil which proves that vaccinations cause a third eye to grow out of the child’s head. Based on that evidence, vaccines should be outlawed.