Shorty's BBQ on South Dixie by Dadeland Mall. Rositas Mexican in Florida City. Oneburger on Alhambra in Coral Gables. Stoupsy's Greek downtown on SE 1st. Churchill's Hideaway Wed. curry night in Little Haiti. Cafe Bali indonesian downtown on 2nd Ave (weekdays only). You don't have to spend more than $10 in any of these places to get tasty treats.
San Loco on South Beach is open late and has a great burrito grande.
Sauces come mild, hot, serious and stupid. I always order mine extra-stupid, which is basically shredded serrano chile.
A hole in the wall on 10 street and Washington (you foodies correct me if I am wrong).
Focusing on the late night portion of your query (which usually ties into the interesting & cheap factors too), some good suggestions in these threads:
Late night, interesting food ->
Late night burger / wings / cuban ->
Top late night dining FL ->
You can try cuban restaurants such as La Carretta(I think the one on Bird Road is open 24 hours) for breakfast you can get cafe con leche, and cuban toast or some of their pastry items for a breakfast under $5.00, and they have great sandwiches for lunch for about $6.00. The
restaurants are a bit cafeteria like but good food. They usually have daily specials with good deals. http://lacarreta.com/ There is also Versailles that is a cuban joint, I believe it's open 24 hours open in Little Havanna too.
If your willing to go up to Fort Lauderdale area such as Hollywood there is La Granja, a Peruvian place with great roast chicken meals for about $3 to $6 (depending upon if a quarter chicken fills you up or not) comes with rice and beans, plus they got a bunch of lunch specials. They also have one that has seafood too, great ceviche for a good price, can't remember exactly how much it costs. Not open late. 2502 N State Rd 7, Hollywood, FL 33021-3205 (954) 963-1945
There is this place called Littlle Saigon on 163rd in North Miami beach that has Pho which is a fairly cheap item, it's open to 1 am. I have to say it's not the best Pho, but it's open late.
If you want a chain place thats open late there is Miami subs they have specials each day for good deals, I usually only eat their pitas and chicken wings though.... they have like 10 or 15 cent wings on Wed and Sun. http://www.miamisubs.com/
Another chain thats open late and is fairly cheap are Ale houses, great beer specials. http://www.alehouseinc.com/
Almost all of the Jamaican/Caribbean Restaurants I've been to are cheap, great for a hearty lunch or dinner, but don't really have dine in, maybe a few tables and plastic wear to eat. Best bests are jerk chicken and Jamaican patty if you've never had it before, I also like oxtail stew, but might be a little exotic. I usually go to Kingston Delight in NMB or Christine's Roti Shop also in North Miami Beach. I've been to others, but can't think of their names.