Eliezer and Robin discussed g somewhat in their debate. I think this question is one that we can do some more research on ourselves. The current hypothesis I'm exploring is that g measures the ability to take in information non-inductively this includes gossip, culture and taught skills.
If g is a real phenomenon then it has to be explained in some form. What is it that some people possess that enables them to do better in a vast range of activities than other people? It seems likely that it is ability to absorb information from the world in some sense.
When I was studying machine learning I was exhorted to put all the information about a problem domain in the learning algorithm, to make it learn efficiently. Now it was supposed to be learning about the world, so it seemed backwards that I needed to tell it about the problem to make it work. And each problem would need different information. So from this view there shouldn't be one factor that makes a system efficient over large numbers of problems. There is however one regularity about information in the world that could be exploited to help solve other problems, there is lots of information generated by humans stored in standard formats. So if this skill increased, you would expect it to have knock on effects in solving lots of other problems.
I'm avoiding calling it social or linguistic information, as that seems to have connotations of solving social and linguistic problems, where as this type of information can help in solving pattern recognition problems. For example learning the concept of a triangle, that it's triangle-ness is invariant to rotation, scaling etc, can help in IQ tests where changes are made to pictures of triangles and you have to spot the commonalities.
There is a sea of useful information out there in linguistic form. Maths and languages are taught linguistically, we memorise times tables and the letters of the alphabet. Even rationality as practised by lesswrong is counted in this class. What happens if you aren't very good at absorbing this vast wealth for some reason? It would be reasonable to expect that you would be less competent in general than your fellow man. Being able to "understand" concepts or formula is closely associated with the lay meaning of intelligence.
So what can we predict about g if this view is correct?
Taking in non-statistical social information is a multi-stage process. You have to do things like parse the sentence or symbols, examine it against your other beliefs for consistency, follow the logical conclusions of the belief and update it and store it so that it can be used in this process for other beliefs. And probably many other steps that I cannot guess at. Any of these might break slightly reducing the efficacy of the whole process. Only the first step I mentioned is unique to human intelligence, the other steps occur if you infer a belief and want to check it against your other knowledge.
So many genes for g might be present and can break, and are probably present in other mammals.
Looking at the genetics of the g, it seems that it is highly heritable in some circumstances and that multiple different genes do have an effect on it. Which is consistent with this hypothesis. We haven't narrowed down the locus' according to that article, I'll try and find something more recent in a while. The review article by Dreary referenced above has a slightly less specific hypothesis.
Bouchard (1997) has proposed that we inherit not intellectual capacity as such; rather, species-typical affective-motivational systems shaped by the environment of evolutionary adaptation that drive both capacity and preferences. Following Hayes (1962), he suggested that manifest intelligence is the demonstration of skills and knowledge accumulated during the experiences created by these affective-motivational systems.
It goes on to mention that the Flynn effect is consistent with this, that society can accumulate better knowledge which then gets passed onto humans which show.
I'm going to do some more reading of references and will update this post as time allows, but I would be interested to know of alternate explanations. Then perhaps we can see which position of the debate the best explanation supports.
Edited to try and clear up confusions - 5th July
IJ Deary, W Johnson… - 2009