Go out and see the sites! BA is a great city and well worth the visit. I would try and store my luggage somewhere - cruise line?, hotel?. If this was problematic, head out to EZE first thing and check your bags and then return for touring. The cab ride on the toll road is about 30 minutes from the center of town - assuming reasonable traffic.
I like AA's flight schedule to and from South America great (I also live in CA and fly AA). I find the night departures in both directions a big plus in terms of not wasting time on planes. The flight durations are long enough - 9+ hours - to have a decent sleep - assuming you sleep well on planes.
Hopefully your flight is via DFW rather than MIA or JFK (unless you want miles). This makes the last leg of your trip short and you're home before 11:00 AM.
You might also consider buying a day pass for the Admiral's Club at EZE - assuming you are not a member or have AA Platinum or Exec Platinum status. The lounge is kind of small but much more comfortable than the general terminal. You can also get some complementary minimal snacks and drinks.
I would also recommend that unless you are flying in business class to bring ear plugs, eye shades, and a good neck pillow on board. These items can make getting sleep in coach much attainable.