The following is based on a chance encounter with a stranger. I don't believe I am violating any ethical lines, but will entertain criticism in that department. I'm not a physician, and nothing in this should be viewed as more than extreme scientific speculation. If the Gene Sherpa or others feel I should be raked over the coals, then get the bonfire going!
I have a young son & so spend time on playgrounds and similar situations. Last fall I had taken him & his cousin to a playground at a park; he had reached his quota of watching his cousin's youth soccer and needed a change-of-pace. Some older kids were there, resulting in gleeful experimentation with extreme G-forces on the merry-go-round.
One of the other parents there was keeping an eye on the events but also tending to a clearly very challenged girl in a wheelchair -- she had various medical gear on the wheelchair and at least one tube. In the course of routine conversation (no, I wasn't prying!) it came out that (a) the girl was 10 and (b) she had already in her young life had multiple organ transplants. Intrigued, I asked what her condition was called (okay, now I'm guilty of prying), and the answer was that as far as any specialist they had consulted knew, this girl was the only known case.
Given my background & interests, it was natural for me to start the mental wheels grinding on genetic speculation. Don't worry, I don't reserve this for strangers! Shortly after my son was born it came out in casual conversation that a relative on his mother's side was colorblind, and so I went into hyperdrive grinding out the probability that my son would be too. Around the time of his first New Year, he was playing with a green ball amongst red ones, and the lightning hit again! Green amongst red! After several trials, his red-green vision powers were good!
Now, such a multi-symptomatic syndrome could have many causes, but suppose it was genetic? Since there was only one known case, traditional genetic mapping would be impossible. But, what might whole-genome sequencing be able to do?
There are many genetic scenarios, but let's narrow it down to three.
First, it could be a simple Mendelian dominant, as is the case with CHARGE, a developmental disorder. For such devastating diseases to be dominant, they must arise from spontaneous mutations.
Second, it could be a simple Mendelian recessive syndrome, but very rare or a novel phenotype of a known one. Depending on the type of mutation, damaged versions of a gene can lead to phenotypes which are not obviously related. For example, some alleles of decapentaplegic in Drosophila are known as Held Out because the wings are always pointed straight away from the body; other alleles led to the official name as the flies have 15 defective appendages.
Third, it could be something else. Interactions between genes, some nasty epigenetic problem, etc. Those will all be pretty much intractable by genome sequencing.
But how much progress might we make with the other two cases? First, suppose we got a complete genome sequence for the affected child. Comparing that sequence vs. the human reference sequence and catalogs of SNPs (which should grow quite large once human whole genome sequencing becomes common), one could attempt to identify all of the unusual variants in the child's genome. Depending on how well the child's ethnic background is represented in the databases, there might be very few and there might be very many. A sizable deletion or inversion might be regarded as a good candidate for a dominant. An unusual variant, particularly a non-synonymous coding SNP or a SNP in a known or suspected genetic control region, might be a candidate -- and if homozygous might be a candidate for a recessive.
Now, suppose we could also get the mother's sequence as well. Now it should be possible to really hammer on the Mendelian dominant hypothesis, as any rare variants found in the mother can be ruled out, since she is unaffected. If you could get the father's DNA as well, then one could really go to town. In particular, that would enable identifying any de novo mutations (those that occurred in one of the parent's germline (ova/sperm) but they don't carry in their somatic (body) cells). It should also allow identifying any funky transmission issues, such as uniparental disomy (the case in which both copies of a genetic region are inherited from the same parent). Finding uniparental disomy might be a foot in the door towards an imprinting hypothesis -- getting two copies of a gene from the same parent can be trouble if the gene is imprinted.
How many candidate mutations & genes might we find with such a fishing expedition? That is the big question, and one which really can only be answered by trying it out. The precise number of rare alleles found is going to depend on the ethnic background of the parents and their relatedness. For example, if the parents were relatives (consanguineous), then there is a higher chance of getting two copies (homozygosing) of a rare variant (no, I'm not the type to pry that deep). If a parent is from an ethnic background that isn't well represented in the databases, then many rare SNPs will be present.
What sort of gene might we be looking for? Probably just about anything. Genes with known developmental roles might be good candidates, or perhaps predicted transcription factors (ala CHARGE), but it could be anything. Particularly difficult to make sense of would be rare SNPs distant from any known gene -- they might be noise, but they could also affect genetic control elements distant from their target gene, a well-known phenomenon.
For the family in question, what is the probability of getting useful medical information? Alas, probably very slim. One can hope for a House-like epiphany which leads to a treatment, but even if a good candidate for the causative gene can be found that is unlikely. Some genetic diseases involving metabolic enzymes can be managed through diet (e.g. PKU), but many others cannot (e.g. Gaucher's). One might also hope for an enzyme-replacement therapy. However, it is quite likely that such a disease would not be in a metabolic enzyme, and might well be in a gene which we really know nothing much about.
So, such a hunt would be for medical edification. Would it be worth it? At the current price of ~$1M/genome, it's hard to see. But at $10K or $1K per genome, it might well be. It probably wouldn't work in many cases, but perhaps if a program was set up to screen by genome sequencing many families with rare genetic (or potentially genetic) disorders, some successes would filter out. We'd certainly learn a lot of find-scale information about human recombination and de novo mutations.