One of the questions worth considering is:
What does it mean to say that a gene is ‘human-specific’? Or more broadly, what does it mean to say that any gene is unique to any particular species or lineage?
Here are two extremes.
The FoxP2 gene is a huge story because it seems to control language ability. Mutations in the gene create language impairments. Perhaps more interestingly, the human version of the gene seems to have been subjected to strong selection. If we grant the prominence of language as a human-specific specialization, then we have a gene that looks like a candidate for membership in a “gene that makes us human” collection.
But the human version of the gene is almost completely identical to non-human versions. It differs from other mammalian versions by just two amino acids. We know that even a single such change can radically alter a protein’s functions and roles, so my point is not that the two-amino-acid difference is biologically unimportant or uninteresting. My point is that the human FoxP2 gene, structurally speaking, is just a slightly tweaked version of a gene that is found in (as far as I know) all vertebrates. I don’t think we should call it a ‘human-specific gene.’
A gene that is utterly unique to a species would be a coding sequence (meaning a DNA sequence that encodes a protein sequence) that existed in that species but not in any form in its closest relatives. I don’t know of any examples of this in humans yet. The best way to get such an utterly unique gene is to have it hop into the genome from somewhere else. This is known to happen, but I don’t think it is known to underlie the formation of any specific unique human coding sequence. That would be the other extreme.
But between those two extremes are some genes found in humans that are very different, structurally, from similar genes in our closest relatives. The gene may exist in some form in our fellow primates, but the human form is much bigger, or much smaller, or perhaps the protein that it makes is a lot different due to changes in splicing or processing.
So while we don’t yet know (as far as I know) of radically new genes in the human genome, we do have some very interesting examples of that last type of new gene. And very intriguingly, the best examples so far are all genes that are involved in brain development. The confluence of “new genes” with the most notable human specialization that there is (cognitive prowess and bloated brains) is pretty darned interesting.
One final comment: I was focused on protein-coding genes in this post, but there are other kinds of genes (depending on how one uses the word), and there are some important genomic elements in humans that may fall into that “utterly unique” category I created above. Here is a somewhat recent review of those (open access). I haven’t looked to see if there are unique lncRNAs or other similar ‘genes’ in humans but it would be really interesting to do that.