Lilly is rather brief in his statement - simply saying that 'Saturn doth joy in that house (twelfth), for natually Saturn is author of mischief'. Which seems to suggest that Saturn enjoys being in the twelfth because it is a house of misfortune and Saturn enjoys causing misfortune.
However, the system of joys is far more ancient than Lilly, who draws on Ptolemy's definitions. It is also clear that the system of joys also predates Ptolemy and that earlier authors had Saturn 'joying' in the fourth. A possible explanation as to why Ptolemy gave Saturn the 'joy' of the twelfth relates to it's sect - that is whether it is a diurnal (day) planet or a nocturnal (night) planet. Ptolemy lists Sun, Jupiter and Saturn as diurnal planets. The Sun and Jupiter are obviously Hot planets ruling Fire signs. However Saturn is a cold melancholic planet, which would seem more naturally introverted and nocturnal. By treating Saturn as a diurnal planet, Ptolemy was recognising that Saturn is better placed where it can do less harm - in the day sky.
By the same reasoning, Mars is assigned as a nocturnal planet, even though it is a 'hot and dry' planet. It tones down its capacity for 'mischief'.
The twelfth and sixth houses in a chart (together with the eighth) are the weakest houses in the horoscope, in the sense that planets in these houses are the least able to act effectively. Saturn joys in the twelfth and Mars joys in the sixth, because here they will find it more difficult to act (and of course when they do act they act in a 'malefic' way. Notice that Saturn in the twelfth is placed according to sect in a day chart and Mars in the sixth is nocturnally placed (again according to it's sect in a day chart).
Now there may well be times in which Saturn is highly dignified and therefore is more the teacher and the father than the mischief maker but Saturn is never a bundle of fun, much better to have it placed where at best it can provide a gentle prod in the right direction rather than where it can be the author of mischief (mainly in the shape of fear, low self esteem, melancholy and depression).
Incidentally, in terms of 'natural' rulerships of houses, from a traditional standpoint, Saturn naturally rules the first and the eighth (planets being assigned in Chaldean order) The 'natural' ruler of the twelfth is Venus.