It is often considered a very bad idea to try to name a discovery (a law, phenomenon or an invention) after yourself.
On the other hand, there are many species names which seem to be a bastardized Latin of the discoverer's name. With gene names, I'm not sure, but I've seen many frivolous gene names, such as those inspired by cartoon characters. I wonder if you could get away with naming a gene after yourself if you wanted to.
In other cases, such as names of synthetic strains and plasmids, it is in fact preferable to use the researcher's name. Off the top of my head, I recall GFAJ-1 - the surrounding controversy aside, given all the work she's done, what is so wrong about her wanting to name it after herself?
However, why is this so? Apparently, it can even advance to an extreme where many years after a discovery has become established and associated with Dr. Jones, when he writes a review article on applications of the Jones Effect he will still carefully avoid acknowledging this name.
Isn't it convenient to simply name something after yourself when no clever acronym exists? Isn't doing the work of the discovery enough to earn the right to name a thing? Doesn't the fact that the discovery is important enough to be published automatically imply that it's important enough to be named after oneself?