If you are looking for authentic Cuban food, you could go to Versaille where the old Cuban men hang out during the afternoons with their cafe cubanos and cigarettes, talking politics and women. Or, you could also visit Puerto Sagua...any time of the day. This restaurant, open 24 hours, has THE most wonderful Cuban food in Miami. Yeah, you can get the traditional dishes like ropa vieja or you can get stupdendous Cuban sandwich. Warning: If you don't have someone to share the Cuban sandwich with, order a Media Noche (midnight snack-size). It's a little more reasonable in size.
Best recommendation for meals here: I always end up ordering the same thing anymore. Cuban sandwich with either the white bean or black bean soups. Puerta Sagua's soups are amazing. Then accompany the meal with a Mamey shake. Delicious. If you want something stronger to drink, you might opt for a Cuban beer (they'll recommend it, anyway), Hatuey. Enjoy! Oh, and this place can get very crowded, especially late at night. It's very popular with the locals.