How to DIY preserve pet DNA today so that it can be used in 20 years
For the impatient, I'm going to say that probably if you only have a regular home freezer, its unlikely that you will be able to do this. If you have access to a -80C or liquid nitrogen storage its possible, but also less likely. 

Cloning an animal typically starts with the transplantation of a nucleus into an egg cell whose nucleus has been removed.  So unless things change drastically in the next 20 years, just purifying DNA and keeping it in a tube of the fridge is not enough - you need to preserve epithelial (skin cells) or some other tissue in a state which can be revived to produce a live and intact cell when you want to do the cloning.  

Its possible that we will have machines that can take DNA from a DNA prep and produce a living cell - this is what we hope to do with the mammoth and other extinct animals from fossil tissue some day.  But whether that will be the case in 20 years is just a guess, whether it is affordable is also just a guess. 

So from what we know now, you will need to preserve a piece of tissue from the animal when it is alive.  I'm not an expert on how to do this, but you might be able to get the skin cells to sort of turn into a cell line and then freeze them for not too much money.  

If you do this, regular freezers are not cold enough to prevent freezer burn.  This is what happens when you put a steak in to the freezer, wrapped in plastic even it will shrivel up and start to dry out as the water in the ice starts to sublime out of the package (the dry air in the freezer basically sucks the water out of the food).  If this happens to your animal tissue, its probably not going to revive.

Scientific labs use -80C freezers and liquid nitrogen storage because the water turns into a glass and all biochemical reactions are basically stopped. (besides drying out, the enzymes like DNAse are still nominally functioning in the cells at -20C and even simple bacterial cells don't live for more than a year at -20C, much less mammalian cells).  For preserving cell lines, liquid nitrogen is much more preferred.  I would say that properly produced cell lines can theoretically revive after indefinite liquid nitrogen storage.  

So that's a quick answer.  Sorry to be a party pooper - things could change quite a bit in the next 20 years, but we just don't know how much.  popping a paw in a baggie or some DNA extract into the freezer might work, but its hard to say for sure. 

As far as the choice of where the DNA comes from in the animal, its true that skin cell lines are often producing imperfect animals - the DNA may be modified in the skin in various ways that cause the animal to be smaller, weaker, or even deformed compared to the donor.  At this time all the protocols I see (and i could be wrong) are skin cells.  I would expect that there is a better tissue to preserve, but that might be just a guess at this point.  Its likely that in the next 20 years the choice of cell line from the donor will change quite a bit as well.