Hey Sir Bugsy, how's tricks?
I agree with you that variant games were what kept the Civ-enthusiasm alive, so if you want to keep things fresh, you need to do different things. Most of the SGs I played were variants and there was always a case of trying to do things that had never been done before. I don't play Civ3 anymore (though looking forward to Civ4) but I do keep track of the SG forum, precisely for games that are unique or done in an interesting manner...in the gaming world, there is little better than a well-run, heavily analysed SG.
However, as you noted, there are very few new ideas coming onto the scene and most games are repeats or copies. Another trend I've noticed, which was anathema a couple of years ago, is that SGs now seem to be starting with everyone rolling up maps and then taking the very best start based on food bonuses, fresh water etc. What happened to the old-fashioned ideal of playing the hand you're dealt?! I remember Random Deity Troublemaking, which I think was the closest SG ever (victory by half a turn), and that game was such fun because we were fighting against the odds the entire game as we took what we were dealt. So for any current SG players interested in my $0.02, take the first start

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As far as creating variants goes, my advice is not to think in game-language terms (units, movement, techs, gold etc) but instead to think thematically, think of an idea, a belief or a goal. Then translate that idea into a game variant. IMO this method produces more convincing variants than simply applying game restrictions for the sake of it and not taking believability or logic into account.
So here's 3 ideas for now:
1) Pontius Pilate and Biggus Dickus Take Control.
I'm sure you all know the scene from The Life Of Brian. Pontius struggling manfully with "welease Woger" and Biggus trying to get his tongue round "thicth theditiouth thribeth from Thaetharea".
The theme for this variant is you can't get involved with anything that begins with "R" or "S". You can't talk to those civs (Rome, Russia, Spain etc), you can't build buildings (recycling plant, research lab, SAM, solar plant etc), you can't build the wonders (SETI, Sistine, Shakespeare's, Smith's, Suffrage, Sun Tzu, Statue of Zeus etc), you can't research the techs (writing, steam power, republic etc) but you CAN buy them (otherwise you'd never progress), you can't train the units (spears, swords, rifles,
settlers *** etc), you cannot use the resources (silks, spices, rubber, saltpeter).
You can iron out the details of exactly what you'd do but the general theme is that Pontius and Biggus cannot get involved in things they can't pwonounce pwopewly, though I'd stick to starting letters only

. You could use any civ (Wome?) or create a custom civ. Since you can't train settlers it would also be a one-built city game and you'd have to conquer or culture-flip to expand. This should be manageable up to deity, emperor would be the optimum level to have a bit of fun. I'd suggest a setup of:
Map: standard, 60% pangaea, hot, dry, 4 billion years.
Civ: Wome or custom
Opponents: Russia, Sumeria, Babylon, Hittites, China, Persia, Greece, normal aggression, raging barbs.
Difficulty: emperor.
2) The Mongol Hordes.
Simple, high level conquest game with the pressure of time. Defeat all other civs, raze their cities to create pasture land for your beloved horses.
Restrictions:
a) must win by conquest.
b) 10CC...no more than 10 cities at ANY point.
c) can only train horse-based military units: chariots, horsemen, keshiks, cavalry. Artillery and naval transport units are allowed. Warriors, spears, archers, swords etc are NOT.
d) must raze all enemy cities.
Map: standard, 60% pangaea, cold, dry, 3 billion years.
Civ: Mongols
Opponents: China, Korea, Russia, Arabs, Germany, France, Spain, maximum aggression, sedentary barbs.
Difficulty: deity.
3)
Doing Deals With Baddies. That was my last game but I never got it off the ground as I had computer problems at the time. See the thread for the variant...run properly, it should be do-able and fun.