An eponym is something named after a person - real or fictional. In medicine eponymous conditions are usually named after physicians who first describe them and, occasionally, after unfortunate patients who happen to suffer from them.
While most eponymous conditions usually have an alternative, non-eponymous name, there are several reasons why eponyms are likely to be around for some time.
Eponyms are convenient. Broadly speaking, all eponymous conditions can be divided into two major groups - ubiquitously common and painfully obscure. The first one includes conditions such as, for instance, Crohn and Alzheimer diseases. Indeed, so common they are that using the identifier "disease" has become optional, and they are often referred to as simply Crohn's and Alzheimer's. These are known by their eponymous names not just to healthcare professionals but to laypeople alike, and for this reason are likely to be referred to as such in the forseeable future.
There is also, in contrast with the previous group, a number of esoteric eponymous conditions with only a handful of cases of each described in the literature thus far. These are likely to stay too, even if for reasons quite opposite to the previous group. Most practicing physicians will most likely never encounter a case of Devriendt syndrome during their career. If they do, however, the eponym provides a convenient point of reference.
Eponyms are, as a rule, shorter than their non-eponymous counterparts. It takes significantly less time (and articulatory proficiency) to define a neoplasm as Evans tumor as opposed to low-grade fibromyxoid sarcoma. While it is considered a standard to include both names in published papers, in everyday practice using just the eponym would suffice to convey all the necessary information.
Finally, there is a small subset of eponymous conditions for which there is no non-eponymous counterpart, Ewing sarcoma being the most notable example.