A 5.5 gallon from PetCo should run about $12. A few pounds of sand (you may have to buy a whole bag but that should be about $10). A
HOB filter can run $10-$30.
live rock will be $3-$4 a pound, and you'll want 5-10lbs depending on the shapes of the rock and space you want to fill. You can buy pre mixed saltwater at the store, or buy
RO or distilled water from the grocery store for around $1 a gallon or so and get your own salt mix. I don't know the pricing on this but these should be relatively cheap.
Now we get to the lights. Assuming you went middle of the road on the filter at $20, got 8lbs of rock at $25, the tank for $12 and salt/water for $20, this could be your budget breaker. What kind of things do you want to keep in the nano? If it's your first then I would recommend just softies and a goby or two. You could go with Compact Fluorescent lighting and support some easy corals, but I doubt you'll find one for $20 (also, I don't know the width of a standard 5.5 gallon.