Well, all teenagers are different, in what they like.
I have 2, and when I bring them to Buenos Aires, they are happiest if I just give em a hundred pesos and let em go. They take the subte to Florida, and buy junk. They go to the malls. They shop at Zara. They buy super panchos and candy bars and caffeinated beverages.
But they also love stuff like Peru Beach- http://peru-beach.com.ar/2009/
windsurfing, kayaking, tennis, and, of course, the above mentioned super panchos. or hamburguesas.
the zoo is great- dont know where your teenager is from, but mine have never seen Carpinchos up close, and they loved that you can buy a bucket of food and feed most of the animals, often right from your hand.
rent bikes in Costanera Sur.
Futbol, if so inclined.
The observatory in Parque Centenario is free on weekend nights.