Per person would be quite easy. For a family, that would be very hard IMHO. But here are things I eat for lunches and I think they are pretty frugal.
A couple scrambled eggs and a piece of fruit or some sliced veggies.
A container of yogurt and 1/2 a sandwich.
A baked potato with diced meat and some shredded cheese.
Grilled cheese sandwich with tomato soup
A cup of steamed rice with a pat of butter and some steamed veggies mixed in, maybe some cheese on top
A cup of noodles with a little jarred sauce or a pat of butter or drizzle olive oil, maybe a sprinkle of cheese and some sliced veggies
A cut up apple with peanut butter or sliced celery with peanut butter and a container of yogurt.
A dozen eggs are about $1.79 here; apples are $1.79 a lb, bananas are .79 a lb, a block of cheese is $1.99 a lb, loaf of bread is $3, bag of pasta is 99 cents, bag of baby carrots is 99 cents, lb of celery is $1.50, container of yogurt is 60 cents, lb of sliced turkey can be $2-$4, etc. etc.
So using a bit of those things would keep a meal to a $1 or less. Granted, these lunches are not fancy and some would say not filling (like my DH, LOL) but for me, they are just fine. Better than a $1 cheeseburger at a fast food drive through!
And leftovers, why those could be even more frugal!