This is actually quite tricky, because some civs have one brilliantly-designed element but are either let down by others, or the uniques have no synergy with one another.
Arabia, to my mind, has the best-designed UA in the game as of BNW - it's a very good way of representing multiple well-known elements of a single long-lived culture, with an emphasis on the trading you'd expect, while in game terms it's an inventive way of using the trade system for indirect benefits.
But the Bazaar, popular and powerful as it is, is a very bad piece of design. It has no synergy with the UA, as it merely duplicates resources and duplicates add no value to a trade route. Markets are something you build everywhere, but the Bazaar's ability scales very badly, being potentially overpowered with the first one or two duplicates and useless afterwards. It has the potential to be ridiculous, but also the potential to be useless depending on what your starting resources are, if you have duplicates, and if the civs you have to trade with have the same resource.
As a civ overall, the Zulu are one of the best designs. They do what they say on the tin, and do it well - Civ IV had an "Impi rush"; with Civ V's unique abilities and more creative unique units the Zulu (coming to the game late when it was already full of warmongers, and having built a lot of fan expectations) bring this back in a way that's been very positively-received and with a wholly unique UU which is strong but not overpowered, and whose ability makes a lot of sense thematically.
Ironically, of the choices below highlighted as bad designs I'd say all (except India) are generally very well-designed in one or more regards:
Sweden - I don't care about the fact that Caroleans are awesome or that a strategically targeted early Industrial war can generate GGs and possibly some allies. The unique units simply do help Sweden's best victory condition (diplomatic) as much at they should.
While I agree with this, and Sweden having two UUs is a weakness given the focus of the UA, most UUs for non-militaristic civs have little synergy with the civ as a whole, so some leeway is permissible. Sweden does have a very nicely-designed UA and a unique niche among diplo victory civs, and the Carolean isn't a bad representation thematically of the real-world formation. The unique cav unit is a letdown, and design-wise looks a bit lazy - take a Khan, reverse its effects.
India - They're supposed to be good at going tall, but unlike the Aztecs, they don't get anything to help this goal except a happiness boost. (a boost that comes with a penalty that generally kills REX) Sure, the UU is essentially a cheaper, faster CB that upgrades to a Knight, but in my experience it isn't anything to write home about. The UB, which comes at the same tech that makes the UU obsolete, doesn't do much either. Because of its design, India is weaker than America, in my opinion.
Agreed with all of this - the key failing with India is simply that, while its uniques aren't bad in isolation (and as a new tourism building the Mughal Fort is probably now pretty good), it has no growth bonuses. The happiness boost doesn't help them go tall particularly, because they aren't going to be growing any faster than anybody else. Most civs can grow tall without running into happiness problems - India is only at an advantage if it has some way of growing taller to make use of its excess happiness (or conversely some Golden Age-type effect that benefits from lots of surplus happiness).
Siam - Like Sweden, I don't feel that this civ is weak, but their uniques simply don't help much for a diplomatic victory.
Not a weakness with Siam, because Siam isn't tailored for a diplomatic victory. It's tailored to make use of city-states - you will quite often (and I prefer to) go for a diplomatic victory because you've already put the effort in to get the CS bonuses, but Siam is not a "diplo civ". It was suggested pre-BNW (since culture victory worked differently) that it was ideal for a culture victory, but Siam is a very solidly-designed civ with a UA that's very interesting to play in a variety of ways and is fully capable of going for any VC. In that regard the Wat still has some synergy since you need culture for every VC, and it can count as a free culture building when going Tradition. The elephant has no relation to anything else, but see above re UUs for non-warmongers.
Songhai - Ever since GnK, the embarkation defense isn't that great, fighting over rivers never really was, and their UB is essentially a weaker version of Egypt's. Their UU is a knight that simply doesn't get penalized for attacking cities. Again, not much to write home about.
Songhai's strength is that it does have synergy. They have a UA which rakes gold from capturing cities and a UU which, as a knight without the city penalty, is the strongest medieval city attacker except Naruesan's Elephant or the Conquistador (both also knight replacements). The Mud Pyramid Mosque is maintenance-free, so has minor synergy with a civ whose modus operandi is "attack, get money, buy more army, attack, get more money, rinse and repeat", but is on a bad tech path for domination. One abiding weakness of the Songhai design is that the Mandelaku has not been fixed following the release of the Spain DLC, since the Conquistador also has the 'no penalty attacking cities' ability and other bonuses besides. But generally this is one of the most synergistic civ designs.
Let's not neglect the Mongols. There are quite a lof of "rampage era" civ's out there, but many focus on early-era bum rushes. All three uniques come together with the Mongols so that they truly are a terror during the middle ages.
The issue with the Mongols is that their UA doesn't really have a lot of synergy with anything and doesn't accomplish a lot. Both UUs are very well-designed, but I mark them down for a UA that doesn't obviously capture who the Mongols were thematically and doesn't help them particularly with any victory condition, including the obvious domination.
Spain and Inca came out together, and whoever dreamed them up really should get an ovation. I think the definition of a good design is that has an implicit (not explicit) strategy and personality. It doesn't just grant a bag of goodies with no strings attached.
Spain's been criticised a lot for being a "roll of the dice" civ - it's going to rise and fall with what happens to be on the map, and I feel this is fairly poor design (see my comments on the Bazaar). A strong design should be one you need to work to exploit, and that plays differently depending on the game environment but is not a slave to the map. The Inca fit this mould - sure, you don't need to do a lot of work to exploit their UA, but they're strong on certain maps without being crippled elsewhere (every map has hills somewhere).
Korea and Ethiopia both have overpowered UA's, but that's not this discussion is about and they do both have cohesive designs (indeed, if they weren't so cohesive then perhaps their UA's wouldn't be so OTT)
Ethiopia is brilliantly designed to capture its stated theme, but it lacks synergy between its UA and its UB.
Korea was also on my shortlist of the best-designed civs (although like Sweden it's another non-warmonger with two UUs).
I'm not sure this can be called good design, because the civ wasn't actually designed this way but came about as a result of both a patched UA and game changes with BNW, but I've been impressed by Austria's design in a recent game.
With changes making city states more valuable as independent entities, even when not going for diplo victory, Austria makes you seriously consider every marriage - you already have to be allies with the CS, so you're in essence exchanging one perk for another, having already had to work to obtain that CS to begin with. You're also unlikely to be taking city-states early game now. This is not an ability you will now automatically click every time you hit 500+ gold and have a willing ally, as it was in G&K's early days, and this speaks to what I say about good design above. When, whether and where you use this is going to be both map-dependent and reliant on the player working for the bonus and targeting the specific CSes you need at a given time.
Then there's the coffee shop. Forget Sweden, Austria has a building that grants +25% GP production that nobody else has. This has been overlooked as far as I can tell, but as of BNW this is a very big deal. Austria is fairly flexible with its victory conditions, but GP boosts now tend to favour cultural victories because you get boosts to three different GP bars.
