There are a few options.
1. You can just let your visa expire, pay a fee at the airport when you leave and hope you don't run into further trouble. Although, this option has been recommended to me often, I don't particularly recommend it. It seems that the Argentinian government has become a bit more stringent of late, and in addition to being fined the next time you leave the country, you may not be allowed back in.
2. Renew your visa for another 90 days at the local immigration office. There's a 300 peso fee for this service.
3. Leave the country for a few days. Argentina borders Uruguay, Chile, Paraguay, Bolivia and Brazil, so you have lots of options. If you're in Buenos Aires or Mendoza, it's easy to just hop on a bus or boat and head across the border. In many cases, a restful weekend drinking wine in Santiago or visiting the market in Montevideo will cost less than the fee to renew.
It's a bit more complicated in Salta, though, because travel between here and just about any other country is likely to be more expensive than the renewal fee.
It was quite the treasure hunt, though, to find the correct place to renew. We first checked with tourist information who sent us to Polica Federal who sent us to the Isicana Language school, which is the place Argentinians go to procure visas to the United States.
After much running and mucking about, we finally located the correct government office.
Migraciones at Maipu 35, just off Caseros.
The people in the Migraciones office are absolutely lovely, very nice and helpful but be prepared for this process to take at least a couple of hours. Bring a book, music and some patience, and you make the the whole thing much easier for yourself.
As soon as you arrive at this small, white painted office, take a number from the front desk. When finally called, you will be interviewed, asked about your profession, where you live, do you have a university education.
You'll need to give photocopies of your passport, all pages, not just the first page. There's a kiosco at the corner of Maipu and Caseros where you can make copies for .15 centavos a page.
Immigration will give you a voucher which you will then take to the bank Banco Nacional de Argentina, corner of Belgrano and Mitre, where you will deposit the 300 peso/person fee directly into a specified account.
UPDATE DECEMBER 10, 2010: You can still pay your voucher (boleta) at the bank on Mitre and Belgrano, but they have opened a new branch at Florida 575, south of San Martin. Much, much faster. I was third in line and the whole process took about 15 minutes. The other bank branch was packed. I would have been there for hours.
You'll return to Maipu 35 with the voucher, copies and other information. There will be forms to sign, and you'll probably wait longer while your paperwork is passed from the one person to the next and finally to the person who will ultimately give stamp your passport with a fresh visa.
There are no extensions to a tourist visa after this, though. Then you must leave the country, if only for a days.