How much energy does Earth surface radiate at night per square meter?
I'm not sure how accurate you need, but beginning NASA's global 398.2 watts per square meter estimate from their energy budget a rounding down based on temperature variation should give a pretty good estimate.    A lot of people have a problem with that number because it's more energy than we get from the sun, but that's because our atmosphere is a kind of blanket that traps heat, which warms the Earth above it's planetary eqilibrium temperature.

It's also (but less so), higher because Earth is warmed from below as well as above.



Diurnal temperature variation is between highest day temperature and coldest night temperature and you're looking for average.  Using that number won't give a correct answer, but using, perhaps 1/2 that number might be in the range and that's over land.   The diurnal temperature variation over water is much lower.

So, 288 Kelvin Earth, say 12 degree variation day to night, so 294 / 282.  Ratio of 1.0425, to the 4th power means 18% more heat is radiated from land during the day than at night.   Adjust the temperature if you think mine is off.    That's 18% x 29% of the surface area.

For oceans, 71% of the surface area the temperature variation is smaller, say 2.5 degrees.   289.25 / 286.75, ratio of 1.0087.   4th power, means the oceans radiate about 3.5% more energy during the day than at night.

(18% x 29%) + (3.5% x 79%) = about 7.7%

So if Earth's surface radiation global average is about 398.2 w/m^2, then the day/night numbers work out to about 7.7% variation between them, or +/- 15.5 degrees, about 413.7 watts/m^2 during the day and about 382.7 at night.    That's a pretty good estimate I think.   I'm open to a better estimate if someone can think of a way to do it.