I wanted to suggest a few variants and ask a few questions as well.
Pacifist. It's probably been thought of before, but what the hey. Obviously, you may never declare war. You may only attack units that are at war with you and inside your borders. You may never attack a peaceful opponent's units inside your borders, but you may eject them. You may never attack their cities. You must accept terms of peace with a rival that declared war on you. Maybe not if he demands a tech or three... You may also never join any alliances or embargos, but mutual defense treaties might be okay. You may not build armies.
I guess it's not much of a variant, the restrictions are probably few if you're used to playing "builder" style. And you won't win any conquest/domination games either.
My second suggested variant is a form of 5CC, where you may never build more than five cities, but a scenario is created so everyone starts with 5 settlers and may never build more. Hopefully this can be done and still allow random map generation. Can it? The idea would be to avoid those annoying border disputes, culture flips, and most of all the pesky AI that tries to send a settler/spearman across your territory. You would also see more use of colonies, which doesn't get used enough.
Questions: when playing One-City Challenge, how do you get those mid-to-late-game resources like rubber or coal that you need to upgrade your riflemen or roads? There appears to be fewer of those on the map than the number of civs, and in the game I'm playing no one has more than one source of coal. Also, when you don't have to worry about corruption/waste, what's going to be the best government? Democracy still?