You may have formed the impression that saving your own seed potatoes (as opposed to true potato seed) is fraught with danger and liable to lay waste to the local economy. Fear not, it can be done, and while it is a bit late in the year for Northern gardeners to undertake, those in the South might welcome the news, and northerners can start to plan for next year.
It is true that simply saving the smaller tubers from your normal harvest is not best practice, because they may well be carrying diseases. Instead, plant a row of potatoes specifically to produce seed tubers. Keep a close eye on them, and remove any that do seem less vigorous or diseased in any way. Quite early in the season, perhaps late July or early August (in the UK) cut the tops off completely. Exact times will vary, so have a gentle root around beneath the bush and confirm that there are some small tubers there, and then maybe leave the plants another week. Removing the tops prevents aphids from delivering the pathogens they transmit to potatoes, and doing so early is a good idea because the aphids accumulate more pathogens as the season progresses. Dig up the tubers, gently again, because the skins might not have set fully yet, and don’t bother trying to get them too clean at this stage. Set them under cover but in good sunlight for 3–4 days, making sure there is good air circulation around them. They skins may well turn green. This is a good thing, as it helps to make the tubers go dormant so that they don’t sprout prematurely. Gently brush any loose soil away, label the potatoes and store them in a cool, dark, frost-free place.
Rebsie had a lovely post about saving her potato diversity some years ago.
A slightly more advanced technique, especially if you didn’t manage to set aside some plants specifically, is to plant saved tubers into a large pot, using a sterile soil mix, as early in the season as you can. When the shoots are about 20 cm tall cut off the top 5 cm. Plant these cuttings into more sterile soil mix and grow them on. They will have rooted within about two weeks and can then be hardened off and planted out into the garden. You can even over-winter potatoes as cuttings if you can give them a frost-free environment.
You can go one step further yet, and produce your own micro-tubers, but that takes a well-equipped kitchen and a modicum of know-how about tissue culture. Or a small laboratory. It isn’t difficult, but nor is it for the faint-hearted. My own feeling, having explored this once before, is that people with an interest in edible biodiversity would beat a path to your door for clean, healthy stock, if only you were permitted to supply it.