Several answers here.
First there is the Citrus Bud Mite, Aceria sheldoni, which causes fingers particularly on lemons and is fairly widespread. Presumably this is some kind of genetic response to the damage caused by the feeding mite.
Buddha's Hand Citron is a selection made in China in the distant past and propagated vegatively since then. I guess it was a one-off mutation but could not naturally reproduce because the fruit lacks any seeds. There is clearly some inbuilt tendency to produce fingers somewhere in the citrus genome, because there are other examples such as Citrus aurantium 'Corniculata'. These are not the result of any pest, or any disease that can be transmitted.

On other than Buddha's Hand, it's due to citrus bud mite.

see UC IPM at
http://www.ipm.ucdavis.edu/PMG/GARDEN/FRUIT/PESTS/citbudmite.html

And here's an article from 1953
http://ucce.ucdavis.edu/files/repositoryfiles/ca707p5-71858.pdf

PS: this is an eriophyid mite. Much different than the more familiar spider mite.

Here is a link that might be useful: citrus bud mite, Aceria sheldoni

Very interesting, thank you! Searching for the corniculata, I ran across the citrus classification page which offered some really interesting information. Now that I think of it, I've also seen on Navel and Valencia oranges in the local markets, raised almost 'dorsal vein' growths which, if their expression was taken even further, might have generated "fingers" like some of the odd raised fingers shown due to mite damage. The citrus bud mite page was quite interesting, thank you! The garden the fingered Eureka came from has one each, mature Eureka and Meyer lemon trees, neither of which has ever produced anything like this one odd piece of fruit. Interesting the mites are more prevalent along the coastal areas as these trees are two mountain ranges in from the coast, nearly fifty miles from the beach, where frosts usually keep insects under control. Thank you! Kim

Here is a link that might be useful: Citrus classification

" I've also seen on Navel and Valencia oranges in the local markets, raised almost 'dorsal vein' growths"

I think you are describing what are generally called chimera fruits. Here, a segment of the fruit reverts to some other variety with different characteristics - colour, peel thickness, rougher texture etc.
Citrus chimeras - not just the fruit - have fascinated scientists for centuries. If you are interested in this, take a look at my webpage about the Bizzaria variety!

Here is a link that might be useful: Bizzaria