It depends on what cravings you happen to have at the moment.
Wanna play as the ultimate D and steal everyone's city states? Go for Austria. Venice is always an option, though inability to fund cities is just unacceptable for me personally.
India is good for a quick, culture-based game. Civ BNW favors tall empires (few cities, lots of citizens) so you won't be lagging behind in anything and the game will be quite fast-paced. Less territory means less maintenance.
Aztecs, Zulu and Huns are great if you just feel like destroying something beautiful. Aztecs will allow you to do it with style, the Zulu have some interesting flanking bonuses and Huns can burn cities twice as fast which eliminates the annoying part of war when you cannot afford to conquer anything because of ruined happiness.
Muslim strategy with a religious civilization is also fun. Inca and Ethiopia allow for huge early faith gains plus religion-based combat bonuses. Pick islam as your religion, denounce everyone who doesn't believe that islam is religion of peace and send your holly warriors to behead them all. You can also pick Arabia for that and hope for a desert start.
On one hand, the French have a quite entertaining unique ability and so does Brazil. Extremely circumstantial and determines your progress from the very beginning but building symbiosis with terrain is quite fun. Something similar could be said for Polynesia, though they can be frustrating if you get an unsuitable map. On the other hand, there is Poland with their versatility. After a little bit of exploring, you can decide on pretty much everything for the future.
Last but not least, there's Spain. Go on a hunt for natural wonders and make sure that all the wonders are belong to us. Small Continents Plus map script is the best for this approach.
The biggest strength of Civilization V is the fact that with theoretically very limited options, you can play millions of completely different games. You'll surely find something for yourself.
