It isn't a very good idea, but that doesn't mean people won't try it. Doing a simple PCR really is pretty easy; I've done it in a hotel ballroom (proctoring a high school science fair sponsored by Invitrogen). Instructions for homebrew thermocyclers are surely out there; a number were published in the early days of PCR. But that doesn't mean getting good results is easy. Sticking to a purely technical level, are Wired's instructions very good?
I'd say no. I suppose I should even register to edit the wiki, but at the moment I'll limit myself to pointing out some of the technical issues that are ignored or glossed over (the material I quote below may well change, since it is a wiki).
The first obvious area is primer design. Wired's instructions are pretty simple
Designing them may be the hardest step. Look up the DNA sequence flanking your genetic marker of interest in a database like dbSNP. Pick a segment that is about 20 bases long and slightly ahead of the marker. That is your forward primer. Pick another 20ish base sequence that is behind the region of DNA that you want to study. Use a web app of your choice to find its reverse complement.
Alas, this will frequently be a recipe for disaster. As for my own qualifications for making that claim I will state that (a) I regularly design PCR amplicons in my professional life and (b) I have a much greater appreciation for my ignorance about how PCR can go awry than the average biologist. Leading the list of pitfalls is designing a primer with too low a Tm -- if those 20 nucleotides are mostly A & T, it won't work well. Second would be if the two primers will anneal to each other; you'll get lots of primer-dimer and little else. Equally bad would be a primer that can prime off itself. Third would be if the primers aren't specific to your targeted region of the genome. Prime off a conserved Alu piece and you are in real trouble.
The really silly part about this advice is that there are free primer design programs all over the internet, and some of the sites will perform nearly all of the checks mentioned above.
The rules for placement are much trickier than suggested. If you are going to sequence (and you might be sequencing heterozygous DNA; see below), then you really need the primers to be at least 50 nucleotides away from what you care about -- there is a front of unincorporated dye which often drops the quality any closer than this.
Even more of a concern is the sequence data itself. Wired makes it sound easy
Once that's done, you can buy sequencing equipment and do it yourself, or send the sample off to any one of many sequencing companies and they will do it for about five dollars.
If you are sequencing uncloned PCR products, then you are sequencing a population. If you are heterozygous for a single nucleotide, that means that nucleotide will read out as a mix -- two overlapping peaks of perhaps half height. A deletion or insertion ("indel") will make the trace "double peaked" from that spot on.
Those are the best case scenarios. If you had poor quality amplification (due to badly designed primers or just a miserable to amplify region), all those truncated PCR products will be in the sequencing mix as well -- further degrading your signal. If your SNP is in a region expanded due to copy number variation, then life is even harder.
Which gets to another point: Wired seems to be ignorant of copy number variants. Their testing recipe certainly won't work there.
The idea of untrained, emotionally involved individuals trying to interpret good genetic data is scary enough (Wired's example of celiac disease, as pointed out over at DNA and You, is a particularly problematic one); scarier is to overlay lots of ambiguity and error due to sloppy amateur technique. Hopefully, few will have the energy & funds to try it.