Question

What body exerts the weakest gravitational force on earth?

Answer 1

Gravitational force depends on mass and (square of) distance.

Explanation:

So the Sun, although it's 150 million km from Earth still exerts a great gravitational force because of its enormous mass.
The Moon is much, much smaller, but relatively close, so it also exerts a considerable gravitational force (and vice versa).

So the body with the weakest gravitational influence should be far away and/or have a small mass (as compared to Earth).