···As for the term "nigger", it has become pretty much the ultimate politically uncorrect term in modern times, but when Mark Twain was writing his wonderful books it was the standard word in the vernacular for most Americans, particularly rural Americans...and both black and white Americans...used when they were speaking of a black person. So Twain had it in his books, naturally, because it was an ordinary term at the time.··· Little Hawk

Moreover, Twain was writing about the past: H Finn was written 1870s but set in the 1830s. As LH rightly points out, Jim refers to himself as 'a nigger', without any particular embarrassment.