As a Tourist, he can stay up to 3 months and I see he is just around that time, but as an Argentinian entering with a foreign passport, I'm 99% sure he can stay up to 180 days. I'm saying this because I have been using my US Passport to enter Argentina for the last 15 years, and they always write 180 days next to the stamp... but that had changed since last year because there are no more "passport stamp" when you enter/exit Argentina.

He can call (011)4317-0234 and ask about that, I'm sure he won't have any issue, worst case he will need to pay a fine when leaving Argentina, like 1.500/3.000 AR$ (less than 6 dollars).

As additional note, tell him to create an account with miArgentina, then make an appointment with RENAPER, that way he can renew his Argentinian Passport and DNI for less than 10 dollars (4.300 pesos), it will be shipped to the Arg. Consulate of his district (LA in his case) and it will take 4 to 6 weeks to get it, this is way cheaper than the cost in US (which is $185 for the passport and $15 for the DNI). I don't use my Argentinian passport, unless I have to visit a Country that require a Visa to my US Passport, like some countries in South America (Bolivia, Paraguay etc).