The prototypical Mary Sue is an original female character in a fanfic who obviously serves as an idealized version of the author mainly for the purpose of Wish Fulfillment. She's exotically beautiful, often having an unusual hair or eye color, and has a similarly cool and exotic name. She's exceptionally talented in an implausibly wide variety of areas, and may possess skills that are rare or nonexistent in the canon setting. She also lacks any realistic, or at least story-relevant, character flaws — either that or her "flaws" are obviously meant to be endearing.
She has an unusual and dramatic Back Story. The canon protagonists are all overwhelmed with admiration for her beauty, wit, courage and other virtues, and are quick to adopt her into their nakama, even characters who are usually antisocial and untrusting; if any character doesn't love her, that character gets an extremely unsympathetic portrayal. She has some sort of especially close relationship to the author's favorite canon character — their love interest, illegitimate child, never-before-mentioned sister, etc. Other than that, the canon characters are quickly reduced to awestruck cheerleaders, watching from the sidelines as Mary Sue outstrips them in their areas of expertise and solves problems that have stymied them for the entire series. (See Common Mary Sue Traits for more detail on any of these cliches.) In other words, the term "Mary Sue" is generally slapped on a character who is important in the story, possesses unusual physical traits, and has an irrelevantly over-skilled or over-idealized nature.
Source: TV Tropes
The stupidity of trying to negate the unerring identification of River Song as a terrible, truly awful product of fanfiction level writing revolves around the argument that Moffat the showrunner of Doctor Who is not using River Song as his author insertion into Who.
Just because a Mary Sue is more convoluted in its connection to the author makes no bones to me.
River Song is the Poochy of the Whoniverse.
It's also an entirely deliberate and apparently highly successful pandering to fans of a certain age (and gender), tending to be American by birth, who like having a shall we say less athletic and young character doing the impossible kewl stuff because it validates in some truly odd and sad way their own again shall we say... sedentary life choices.
I seriously couldn't like this character less if she was comprised of equal amounts of shit and maggots.