Two syllables, day-mon. This is the correct English pronunciation of the Latin word. The OS usage comes from Maxwell's daemon:
http://www.takeourword.com/TOW146/page4.html
People who say "dee-mon" are probably 14-year-old Satanists in disguise.
If you want the etymology for it, "demon" is the modern spelling of what was essentially the same word. A lot of "ae" in Latin became "e" in English (prae- to pre-, encyclopaedia to encyclopedia). It was originally pronounced like "eye" and corresponds to αι in Greek. Phonological drift and Christianization caused the new spelling {demon) to imply something evil, so classicists in the 19th century resurrected the ancient Latin spelling with its original meaning: merely a spirit.