How would you convince somebody that the momentum of an isolated system is constant?

The easiest and most effective way to convince him is by practical experiment.

For an isolated body, the total momentum should be conserved, this means that:
total momentum before collision = total momentum after collsion
M1V1 + M2V2 = M1V1' + M2V2'

So, get to masses and record their initial velocities. Then let these two masses collide and measure their final velocities.
Substitute with the masses and the recorded velocities in the above equations to verify the law.

Another way to convince him is by deriving the above equation from Newtons's second law.