"Dee-doe" comes from the rules for pronouncing Latin and from the rules for transliterating Greek into Latin before pronouncing it in Latin.
The most likely source for "DYE-doe" is Anglicization of the name in the 1594 play by Marlowe (similar to the angilicization of "Montague" and "Capulet" in Shakespeare).
Since the language is a bit dead, and most of its contemporaries other than Latin and Greek didn't have symbols for vowels, the "real" answer is pretty much impossible to reach confidently. But I'd expect the Latin spelling of the name would probably try to be phonetically accurate, and the English long-I sound is, I think, not very common around the Mediterranean before much more recent times. So if I were a betting man my money would be on "DEE-doe".