I came across this problem in a test and have been able to come up with a solution however I am unsure if it is correct.
I started by building a cube of twice the initial dimensions to bring point P to the centre of a larger cube. This was to bring an element of symmetry to the figure. Since, each cube contributes potential V to the centre, the net gravitational potential will be 8V since there are 8 cubes. Now I wanted to find the Gravitational Potential at the centre of my original cube. So I considered an imaginary solid cube inside the larger one such that point P lies at the centre of both the cubes i.e the larger and smaller one. Now, I found the factor by which the gravitational potential had increased at the centre. Since, gravitational potential is directly proportional to the mass of the particle and inversely proportional to the distance, I considered the fact that on average, particles on the inner imaginary solid cube would be at a distance twice of that between particles on the outer larger sphere and point P. Since the mass of the larger cube is 8 times the mass of the smaller one, I concluded that the potential at point P due to a cube with same dimensions as the original cube would be 2V. Therefore the potential due to the remaining parts of the larger cube would be 6V. If we divide the remaining part of the cube providing a potential of 6V into 8 symmetrical parts, they resemble what the problem asks us to find. Thus, the answer came out to be 3V/4. However I recently received a hint that this was incorrect and have not been able to conclude why.