The weight of an object would be minimum when it is placed at the:

The weight of a body is W = mg, where m is its (constant) mass and g is the acceleration due to gravity at that location. Since the mass does not change, the weight is smallest wherever g is smallest.

As we move from the Earth's surface towards its centre, g decreases steadily, and exactly at the centre of the Earth g = 0. This happens because the gravitational pull of the surrounding mass acts equally in all directions and cancels out, leaving zero net force on the body.

Therefore the weight at the centre of the Earth is W = m x 0 = 0, which is the smallest weight possible. By contrast, g is greatest at the poles (so weight is maximum there), and at the equator g is slightly less than at the poles because of the Earth's rotation and equatorial bulge - but neither value is as small as the zero value at the centre.

Hence the weight of the object is minimum at the centre of the Earth.