A
unit which would probably have
no use whatsoever.
A solid angle based on degrees, the degree equivalent of a steradian, would be the solid angle formed by rotating the distance covered by a degree around its center. While the arc length of a degree is pi/180 times the radius, the area of a stedegree would be (pi^2)/32400 times the radius squared. Since there are 4pi steradians in a sphere, there would be 129600/pi stedegrees in a sphere.
Since the whole point of degrees is to have an integral and easily-divisible number of them in a circle, you can now see why stedegrees would be patently useless.
Perhaps a better unit would be a stegradian, defined as 1/100 of the surface area of a sphere. This would not correlate with the size of a gradian at all, but it would with the purpose of the unit.