In popular fiction, savant skills and autism are almost synonymous. Portraying a character as a savant has become a way of driving home the fact that the character is autistic. The savant archetype is glaringly problematic because of the cultural baggage associated with idea of the "savant," because of the roles that autistic savants are relegated to in fiction, and because the stereotype of the autistic savant is enormously misleading. While "autism" and "savant" may go hand-in-hand in the popular consciousness, the vast majority of autists are not savants and not all people who have savant skills are autistic.
The trope of the autistic savant can clearly be traced back to Dustin Hoffman's portrayal of Raymond Babbitt in Rain Man. In that film, Raymond's savant skills are a significant part of the plot in that they give Raymond's brother, Charlie (Tom Cruise) a reason to exploit him. Charlie is the film's actual protagonist and the plot centers on his character's trajectory from self-centeredness to empathy. Of course the film was tremendously successful and remains influential, and Dustin Hoffman's performance became a blueprint for most autistic characters in popular media. (Fun fact: Kim Peak, the man on whom Hoffman's character was based, was not actually autistic.)
It is difficult to measure the actual prevalence of savant skills because the definition of what qualifies as a truly "outstanding skill" is itself difficult to pin down. The generally accepted estimate as to the percentage of autists who might be termed "autistic savants" is about 10%. The article that I linked to describes savant skills as "islands of genius" in otherwise (often severely) impaired individuals, and this perspective is key to the cultural fascination with savants. That people who are otherwise perceived as profoundly "damaged," and often as "helpless" because they may be dependent on caregivers, could possess high levels of skill and talent is an irresistible contradiction. This apparent contradiction is usually looked upon as an intriguing anomaly and often as tragic, since savants manage to possess areas of talent "despite" their "unfortunate" disability.
The savant is never really admired in the way that the prodigy—the non-disabled analog to the savant—is often looked upon with admiration and even envy. And while the prodigy is often portrayed as putting in long hours of practice in addition to having tremendous innate ability, savants are never portrayed as having to work at, or as being able to relish, their talents. Even within the medical community, savants are generally considered unable to really grasp the "meaning" of the huge amounts of information that they are able to gather and recall. As Stuart Murray points out in Representing Autism, the term was originally "idiot savant." That "idiot" component, with all of its connotations, remains, even if unspoken. People with cognitive and intellectual disabilities are perceived as generally incapable of the kind of self-awareness and insight that our culture associates with the very idea of being human.
Despite its inaccuracy, and the fact that it reinforces the perception of autists and disabled people in general as subhuman, the savant archetype endures in popular representations of autism. I imagine that a significant contributing factor in its continued use is that it saves writers a lot of work. A particular kind of savant skill can easily become a major plot point, as in Rain Man, Mercury Rising, and, sadly, The Girl With the Dragon Tattoo. Making an autistic character into a savant justifies using that character as a plot device rather than actually taking the time to write a fully-developed, multi-faceted autistic character. Since savantism has become a kind of shorthand for autism, writers can get away with failing to portray autism realistically. They can skimp on or omit the cognitive patterns and sensory issues that autistic readers and viewers could recognize and relate to. Constructing a character whose external behavior and inner life are both recognizable as autistic would require research and actual engagement with autistic people. The problem is, in part, that it is still not universally understood that autistic people have inner lives at all.