Well I'm from Argentina and there's a lot of music I'd like to share. You see, the most "national music" we have here is folklore and tango. Buuut, almost noone listens to this kind of music nowadays, so I usually recommend what I use to listen to. For example, I'd recommend bands from National Rock (such as Los Redondos or La Renga. Oh, one band I know that's good is Catupecu Machu XD), they're for me the best bands xd. Also, if I talk about folklore, I'd suggest to listen to old groups from the last century, because I don't like "modern" folklore. In tango there are many artists known in the world, but the best for me is Julio Sosa XD.
If I had to recommend specific songs, I invite you to listen "La Razón que te Demora" , "Motor Psico" or "Origen Extremo". I think you'll like more those than old songs from folklore and tango XD