The weight of a an object is the force exerted on the object by the gravitational pull of the Earth.
If the mass of the Earth is taken as Me, the radius of the Earth is Re; we get the gravitational force between the Earth and an object of mass m as
F = G*Me*m/ Re^2
We have G*Me/Re^2 = 9.81
For the force to become half while G, Me and m are constant, the distance of the object from the center of the Earth has to be increased.
Let the distance be D
G*Me/ (Re + D) ^2 = 9.81 / 2
=> 9.81* Re^2 / (Re + D) ^2 = 9.81 / 2
=> (Re + D) ^2 = 2* Re ^2
=> Re + D = sqrt 2 * Re
Taking the radius of the Earth as 6300 km
=> D = Re*(sqrt 2 - 1)
=> D = 6300(sqrt 2 - 1)
=> D = 2600 (approximately)
The weight is half approximately 2600 km away from the surface.