Every casino has a 24/7 diner/cafe type place that offers quick eats. For the Golden Nugget, that place is called "The Grille". And the creative name reflects the establishment - this is no 24/7 wait service dive diner with charm. It's quick counter-service style with a rather basic menu.
Nonetheless, what I ordered was pretty good. The breakfast burrito, stuffed with chorizo, potato, and egg, is ample and filling for $6. Many cite the slow service here, but I didn't have an issue with the service. There are better places for late night eats on Fremont St, but this isn't bad if you are staying at the Nugget and want something quick and close.