Welcome to FT. I see this is your first posting.
You did not clarify what is a stopover, how many hours and at what time you are arriving. Remember that Buenos Aires has 2 airport: EZE is about 1/2 an hour away from the Center of town and Aeroparque is in the center of town.

I am not sure if you are familiar with the site but we have a great search option. A couple of months ago we had this discussion and I think many of us recommended against staying near the airport.
We felt that even if you are here for a few hours, Buenos Aires has a thriving night life with great food, music, tango, dance and bars that it is a shame to miss.
I think you should fly in, take a US$20-25 cab or car service into town, enjoy a great dinner, have a couple of glasses of excellent wine and then leave in the morning.
I usually rent an apartment when I come to Buenos Aires but once I stayed in the Kempinsky Recoleta Hotel. They have several hotels around town and even though it is a little dated, the service is very nice and the location could not be better. Plan on spending US$100 for a nice hotel, 200 for really nice and 300 for real luxury.
Gaucho, care to comment.