Regarding the use of eponyms in medical nomenclature, I found an interesting paper on the (American / USian / Great Satanian / Gringo) National Institutes Of Health website. It says that there is widespread inconsistent and arbitrary variation between the possessive (e.g. Down's Hodgkin's Alzheimer's Wilson's etc) and non-possessive (e.g. Down) forms. Concentrating on Down/Down's, it notes that the possessive form is more common in European medical publications and the n-p form more common in US publications, but that over time the n-p form seems to be gaining.
The authors assert that "Because of linguistic simplicity and technical advantages, the non-possessive form should be used uniformly worldwide" and cite ease of word searching in databases as a reason.
I have a number of problems with this, due, I will freely admit, in part to a dislike of America telling the rest of the world how to use words, a scepticism about the enforcibility of such a fatwah against the possessive form, and a sentimental attachment to my UK/European mindset. I concede that this may well be irrational and Luddite, and clear proof that I am a far-left Euro-weenie, but there it is.
Regarding the difficulty of word searching, I typed "Down Syndrome" into Google and got around 4,000,000 hits, whereas "Down's Syndrome" got me 876,000. In each case I used quotes around the search term as you see here.
When I searched for "Cushing's Disease" I got 201,000 hits, but when I typed "Cushing Disease" I got 195,000 hits, but -- interestingly -- it looks like they mostly lead to pages where the disease name is actually spelt with an apostrophe and an 's'. I have Google Suggest (which, as you type, offers a dropdown list of suggestions based on frequently entered terms) enabled and it showed me that plenty of people type "Cushing Disease" or "Cushing Syndrome". One result was "symtons of cushing diease" [sic] which is hardly evidence of any particular trend.