South America is such a diverse continent that it's difficult to imagine that you would let the cheapest airfare dictate your destination, especially for a trip of six to eight weeks in length where the airfare won't end up being the biggest component of the overall cost.
Generally you'd expect that the cheaper airfares from either NYC or the Bay Area would be to the northern part of South America, probably Colombia, Ecuador or Peru (I'd avoid Venezuela and the Guyanas for reasons of security). For these destinations you could check for cheap fares on AA, Copa (hub in Panama City and codeshares with CO), LAN and perhaps Avianca (for Colombia specifically). Try out the following website:
If security is a big issue for you, Chile is the only place in SA that I know where you can move about with almost as few security concerns as in North America. That said, single women can and do travel widely throughout the continent and if you take the normal precautions you'll likely be ok. It certainly helps if you speak the language(s).
As for me I'm a big fan of Peru. It's a very diverse country with a fascinating history and a lot to see and do, not to mention great food, and it can be done on a budget. You could easily spend two months there exploring the coast, highlands and rain forest, as well as sites of historical and cultural interest like Machu Picchu.
As cost appears to be a big issue with your trip, note that if you fly internationally into SCL and into EZE with a US passport, you will pay $131 for each country. If you enter Chile or Argentina at other points, you will not pay the fee. If you are planning to visit Brazil on the trip, you will need a Visa which costs about the same (no matter where/how you get there). Most places (if not all) now have all departure taxes included in the ticket, but maybe there are a few that don't, which can be another unexpected expense.
Sometimes the LAN website has some appealing sale fares. From LAX/JFK/MIA you can get to many points in South America with no more than 1 stop. (And via LIM you can fly into IQQ (north of Chile) or COR (Argentina) where you won't pay the entry fee, or via SCL you can fly to MDZ which is another way to get to Argentina w/o fee.
As far as safety, Chile is the safest place in South America. Note that May/June may have some marginal weather, especially as you travel further south. Don't plan on any Patagonian adventures.
For domestic segments within countries, it is often the best deal as Argentina and Peru have residency restrictions on the lowest fares, and Chile has point-of-sale restrictions on the lowest fares. For international round trips, you may find cheaper (but don't expect deals on one way tix).
On the LAN website, you can book them online - you provide your international reservation #. I believe it costs a bit less if you come in on LAN v. the other OW carriers.