I am summoned.
I'm not entirely certain what affects ringed-ness. Rings seem to form under three conditions: first, if a moon is calculated to be created within the Roche limit of a parent body, it becomes a ring instead. Second, if some horrible disaster (like a rogue planet smashing into a moon) is calculated to have happened in the system's prehistory, a ring will form from the debris (I believe this is what's happened in situations where we find large thin rings forming outside of the orbits of the innermost moons). Third, in a system which is calculated to have lots of dust and debris, the planets sweep up some of that debris and form rings.
I have put that third rule in place simply because rings aren't very probable, but far too often, you can enter a system and find virtually every world, big and small, has a set of rings around it. As a general rule, younger systems would be more likely to have lots of dust and debris, so are more likely to have rings; this bears out my qualitative observation that these full'o'rings sytsems tend to be around large, young stars. Young, but not too young; I've noticed many protostars don't tend to form rings around their planets; this may have something to do with protostar planets not tending to have many moons, and thus fewer rings forming under rule 1.
Note that if this theory is true, there are some competing algorithms. For example, "collisions" are more likely to have happened in older systems, since they've had a longer history and thus more chance of such events happening; this would counter the "younger is better" hypothesis of rule 3.
TLDR: it's not really random, but the rulebook is arcane and self-contradictory, so the results are effectively random.