For amusement, someone can go to various websites and say witty things like, "Hey Obamacare supporters, keep your laws off my body." And this is somewhat amusing. Not productive, but amusing.
An alternate idea may be to comment on such websites, and rather than counter others' stupid thoughts, you could just point out what fallacy they are committing.
Some idiot could claim X and you'd reply: "straw man," "ad hominem," fallacy of the anecdote," "projection," "questionable source," and so on.
Actually, you may not need "and so on," most people who argue things on the internet commit any and all of those listed errors.
It would be kinda funny:
Idiot: "blah, blah, blah"
me: "straw man"
Idiot: "Yeah, well you're a racist."
Me: "ad hominem"
Idiot: "You're only repeating rethuglican talking points."
and so on...