Well, if we get a complete overhaul, I would want to have something like
this. Of course, that'd also need a few tweaks and rebalancement to the vanilla game, i.e. a complete overhaul of the navies.
Generally speaking though, they should look at a) those civs that do poorly in the hands of the AI and b) those that have lacking synergy. The A group would probably work well with just some little changes to the personality of the leaders (also, make the early warmongers go first after city states and neighbours).
For the B group, I see those candidates: Ethiopia (Tall and Religion don't mix well so far, but that can change in BNW), Byzantine (needs a religious bonus), Carthage and Byzantine (two classical civs that are very similar (naval+horseman UU)), Songhai (Needs to go for religion to get culture bonus?), Germany (works fine gameplay wise, just not realistic, would need a complete overhaul...*), Celts (just a minor bonus: ease the

requirements and add jungles and swamps to forests), Sweden (complete mess...), America (the UA bonus is a good one, but underappreciated, should get sthg for early rapid expansion), Spain (All or nothing and free

is bad, I'd rather combine the NW extra yields with a bonus against city states who often hog those wonders after all). And so many more
But those are completely unrealistic. A few easy changes I can see them doing:
- Austria's UB gets a focus on the Great Works People (Artists, Writers, Musicians)
- Arabia's UA goes from gold for trade links to something with trade routes
- And that's all. I can't see them doing any big changes..., maybe the Dutch UA since it shares a name with a National Wonder...
*My proposal: Make Scientists equal to Engineers and the other way around, allowing you to beeline up both sides of the tech tree and not miss out on

or

Other proposal: One 'elite' unit per unit line that gets special promotions and is free upon research.