There are actually two problems here. It isn't very difficult to make Ozone -- it is one of the more common industrial pollutants and may be responsible for increased rate of lung cancer in high latitude cities. However, the quantity needed is substantial -- Ozone is being lost by a catalytic process in which the pollutant (C Cl₂ F₂) is not destroyed -- so that a small quantity of this material can reduce a large quantity of Ozone.
The other issue is getting the ozone into the upper atmosphere. Ozone is very highly reactive, and has a relatively short lifetime in the troposphere -- and is indeed heavier than normal air. On the other hand -- it might be possible to introduce a competitive catalytic agent which creates ozone to the upper atmosphere... One still must keep the scale of this process in mind. Millions of tons of Freon are released each year. One possibility for dealing with such scales is creation of a suitable biological organism which can multiply on its own accord.
This has already been done for combating oil-spills, however, the key issue is subsequent control. So -- there might be ways -- but the issue is are the cures worse than the disease?

There are several ways to produce ozone. The most common one is passing an electric discharge through a tube filled with oxygen, and the reaction

3O₂(g) + energy -----> 2O₃(g)
will take place.
Now, even if we can produce ozone in the lab or even in industrial quantities, we as a planet lack the technological capability to produce any gas that would make a significant change in the planet's atmosphere. It is way too big for human endeavors at the current level of technical development.
But that does not mean it is not possible, for things that used to be way beyond the capabilities of humankind only fifty years ago are now commonplace. Here we can only wait and see.
An important additional remark is that the ozone "hole" is not actually a hole, something set there and stable.Scientist use that term to refer to a seasonal variation of the concentration of ozone in the upper layers of the atmosphere. We can say that the ozone layer gets thinner during the winter and almost regains its normal thickness during the spring. This effect was discovered in 1985

There are lots of ways to create ozone - the problem is to get the ozone to the right place to fill the hole in the ozone layer of the upper atmosphere. Too much ozone created at ground level is a major contributor to smog. It would be too expensive to make ozone down here and try to transport it to the upper atmosphere over Antarctica. If some ozone gets produced naturally each year, would it be possible to do something to enhance this natural process, or to reduce the processes that destroy the upper-atmospheric ozone?

I believe the problem is not that there is not enough ozone produced in the upper atmosphere but that the chlorofluorocarbons (CFCs) that have made their way up into the upper atmosphere are very long lived (they are meant to be inert for use on the ground) and disrupt the process that creates the ozone that protects us from damaging solar radiation. We've done a lot to curtail use of CFCs so the situation should improve, albeit slowly.
My guess is that it would take a lot of energy and effort to make additional ozone in the upper atmosphere and spread it around. I think it would also have to be done continuously as long as the problem exists. If the "ozone hole" problem ever became bad enough to try adding more ozone, how do you think we'd go about doing it? I wonder how much ozone would have to be made in order to fill the hole.

That's a great question. The funny thing about the ozone in the atmosphere is that sunlight and other natural things continuously break it down, even before man-made chemicals were ever introduced into the atmosphere. It gets broken down and then re-synthesized over and over again. You see, ozone is really really labile (do you
know what that means?). If you look up labile, you will find that ozone is very delicate, in a way, and is always being destroyed and then re-made.
So, the problem with the man-made chemicals in the air (called CFCs for short) is that they speed up the process of breaking ozone down, but they don't speed up the process of making ozone.
Eventually however, a long as we stop using CFCs it will all be o.k. It is not a permanent problem. Still, the idea of filling big balloons with ozone and sending them up into the ozone layer is pretty good. There are, however, some problems. By the time you get to the ozone layer (several hours), most of the ozone will be gone.
Two ozones can combine to form three oxygens, and that is what happens. Still, we may be able to generate enough ozone (electrically) to theoretically supplement the layer. You've got a good idea about replacing the ozone. But, since we know that it will replace itself over time, it makes sense to solve the problem by stopping the emission of CFC into the air.