I'm sure there's a Greek pronunciation AND an anglicized one. I'm guessing on the English version, en-DIH-mee-uhn (?). And the Greek, en-di-MEE-uhn (also, ?).
Hey, for interest, in French it's pronounced Aw(ng)-DEE-MEE-aw(ng). I like it! The "nickname" would be, Endime, pronounced Aw(ng)-deem, or, Dimi pronounced, dee-mee which is like a French version of "Jimmy". An alternative spelling using French pronunciation is, Endymione, which would make the last "n" say it's sound -- pronounced Aw(ng)-DEE-MEE-ohn.
https://en.wiktionary.org/wiki/%E1%BC%98%CE%BD%CE%B4%CF%85%CE%BC%CE%AF%CF%89%CE%BD