My speculation is that Twain chose to set his story in that time for two reasons:

• It allowed him to set the story in the region of the United States that he knew best.  He knew the river, he knew the dialect, it made sense to set it there.
• Within that region, I imagine that he wanted to find an issue that would very starkly illustrate Huck's distrust of civilized behavior.  I think that slavery, more than any other issue of the 19th century as a whole, showed that society could believe in and condone things that ought to have gone against people's consciences.