What natural phenomena increase stratospheric ozone?
Like BarocliniCplusplus said, ozone is generated when you have a single free oxygen atom. This can come from photolysis of O2 in the upper stratosphere, or photolysis of ozone itself, or photolysis of NO2 in the troposphere. The stratosphere gets its NO2 from the breakdown of N2O by other radical species. But NO and NO2 both react with ozone, typically destroying it.

So if you want a natural process that would enhance ozone, I would look for a process that reduces the amount of N2O (or other ozone-depleting species) reaching the stratosphere. N2O is produced by bacteria in soils, so anything that suppresses that production -- especially in the tropics -- would enhance stratospheric ozone. Carbonyl sulfide (OCS) is another species that only photolyzes in the stratosphere and depletes ozone; its concentrations would similarly decrease if there were a decrease in transfer of it from ocean to atmosphere (it's a product of plankton). You could also have a meteorological effect that suppresses deep convection in the tropics, though I can't offhand think of what this would be.

If you're interested in understanding the specifics a bit more (I'm not great at summarizing all of the cycles involved), I recommend checking out http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap10.html. Daniel Jacob is a pretty influential atmospheric chemist, and he put his whole book online.

NB: NO2 destroys ozone in the stratosphere, but typically enhances ozone in the troposphere. This is because the troposphere has much, much higher concentrations of organic species and the hydroxyl (OH) radical. Instead of depleting O3, the NO produced when NO2 photolyzes can react with HO2 to re-form NO2 and OH, so the ozone-depleting nature of NO is reduced. Since that process re-forms OH, OH can attack organic species and leave the H radical, which then pulls in O2 to form HO2, which will later end up producing an odd oxygen and thus form ozone. (Sorry if this is a bit rambly. I haven't tried to summarize all of this before, so let me know what things are confusing.)