My ideas are probably ridiculous but here they are anyway:
Byzantium: Coastal Start Bias.
Seriously, that's all the tweaking I think they need. Byzantium is the ultimate jack-of-all-trades civ, with a UU that literally lets you pick off the menu for your bonus depending on your situation, but you have to make Stonehenge a priority and that's just going to be the way it is, and at all but the lower difficulties you're going to have to forego the GL in order to get it. What a coastal start bias would do, however, is make sure that you have full use of your Dromond (the most interesting aspect of the Byzantines, IMHO) and also let you explore early on to find the goodie huts that the other civs aren't going to be fighting over so much. That can give you a leg-up on the religion-founding and you can use your UUs in the era they were designed for. Giving Byzantium an innate faith bonus would be overkill, in my estimation.
Dutch: make sure the Sea Beggar has a special benefit related to attacking trade routes.
Perhaps it can capture cargo ships as part of its "prize ship" promotion, and/or perhaps it just appears to all other civs as a Privateer with no flag. I don't know, exactly, but the new mechanics should allow for some fun with piracy.
Rome: Perhaps the Legion could have the option to use the roads they've built to create an international trade route of their attack is failing.
Spain: I've never been a fan of their UA, but should it continue, being the first to discover a natural wonder should have a positive effect on their Great Works.
More ideas when I've got them.
Well, i would put it quite differently. May I give my take on the Civilizations you mentioned?
Byzantines: A civilization that absolutely needs to build a wonder in order to use their UA is a bad civilization.
And Byzantium is incredibly far away from being the abolute jack of all trades: it has a virtually inexisting UA if you want to use any of their units effectively (unless you are playing low-difficilty and can grab a decent amount of those cathrapacts before the AI spam you with pikemen, since in higher settings it's nearly impossible).
All it's bonuses are compressed upon a single era - and one of the worst eras for that, BTW. And not a single UU ability of the Cathrapact or Dromon remain upon promotion. Tis' just a civilization that burn all it's flavor on the classical era - that is, if you don't push for religion. And (even if just sometimes) when pushing for religion, you may FAIL.
Playing Byzantium is just frustrating.
(And I don't get what's special about Byzantium exploration, since you mentioned it. Building Scouts take away time better spent on shrines/Stonehenge... Unless you wanna gamble and reroll the map if no religious ruins are found. Not a very compelling strategy, methinks)
My ideas for Byzantium are simple, too. To make them playable, you needn't make them gambreaking.
(1) Give them a small bonus ir order to get their pantheon fast.
or (2) Give them another entry on their UA that interact with religion founded by other players, in case they aren't able to found their own (like conquering Holy Cities or taking advantage of other religion Founder Bonuses, idk)
and/or
(3) Make the cathrapact a knight replacement
Dutch: I'd rather they got a bonus on trading, like plus 1 happiness for each sea trade route (that would interact with their trading bonus already). The pirate thing is what I hope the Ottomans get.
Rome: Rome's good as it is, even though it's incredibly boring to play as. I'm hoping for some overhaul too, but I don't think it will happen. Even a National wonder building bonus would be welcome.
Someone suggested, some time ago, that they get +1 production in their capital for each city in the empire, instead of their actual UA - and I think it would lead to an interesting playing strategy.
Spain: No comments. I think they need to be rebuilt from scratch.
An idea I had some time ago was a Unique National Wonder for Spain - an inquisition citadel, built with faith, that would make inquisitors cheaper on the city it was built (and yield lots of faith and culture, of course). And frankly, I wouldn't miss either of their UU's that much. I'm having delusional hopes already that the Conquistador go as an UU in the new exploration SP tree and Spain gets something else in it's place.