Wow, never found this question before on Travel.SE. I assure you it's a low limit, and I'll now explain the reasoning behind these limits.
The same limit applies to most bungy jump places - including where I did it, in Taupo, New Zealand. Bungy.co.nz shows some sample minimum weights that differ just in NZ - from 35kg to 45kg.
One thing to take into account is the elasticity of the rope. It can be hard or soft, but the more it stretches, the more it'll absorb your fall and you'll have less force applied to your body at the end of the line. This is why they use 'stretchy' ropes, rather than your garden-variety twine rope, for example. That would hurt like hell.
However, if your body isn't heavy enough to stretch the cord, it becomes somewhat like a garden rope, not absorbing some of the force and dampening your fall. The whiplash and jolt on your body would be severe. This is the reason for minimum weights.
What some companies have taken to doing is 'sandbagging', where they add extra weight to your drop, allowing you to meet the minimum weight. Most places also adjust the ropes depending on your weight anyway, to give you just the right amount of 'spring' and depending on whether you want to dunk into the water, for example.
If someone you know is below the weight restriction, and wants to jump, it may be worth phoning beforehand and asking whether sandbagging could be an option.