To update:

My AA SFO-JFK flight was on time. The JFK-EZE plane mech'ed and was replaced by an inbound 777 from LHR. We ended up departing around 2.5 hrs late and touched down at EZE at 11:30am. We went through immigration and customs, exchanged money at Global Exchange (yes, I know they have horrible rates but we were in a rush!), and took a remise to AEP.

We arrived at AEP around 1:20pm for a 1:35pm LAN flight. Needless to say, we did not make that flight. We bought a walkup fare to AEP-IGR on Aerolineas Argentinas for $117USD that departs 1 hr later. Then we got a refund for the AEP-IGR portion of our LAN roundtrip ticket (less $30).

So we got to IGR on the same day. On the way back IGR-AEP, we had lots of problems at LAN checkin counter because of the refund for the AEP-IGR segment. The agent had to go to the back office for more directions, and finally after around 20 minutes or so, she issued boarding passes (with the same ticket number).

The lessons I learned:
1. leave tons of time between connections in case something goes wrong
2. refundable tickets are nice if you can afford them
3. travel insurance is especially helpful if you book connecting flights on two separate airlines (it will cover my $30 non-refundable fee).
4. run run run, even though you think you can't make it (well, this was only because my EZE-MIA arrived 2 hrs late and I had to make it through immigration, customs, security in less than one hour to make my MIA-SFO flight).