It's a rather difficult question. Of course, both pronunciations are correct today—I think you'd be hard-pressed to find a dictionary that would disagree with that. So, the individual you encountered was wrong about "ī-thər" (I assume you mean the pronunciation that's like "EYE-ther") being wrong. But this question is about the history, and the word has been spelled (and presumably pronounced) in a variety of ways across the years. The OED lists the following forms:
OE ǽg-, œ́g-, éghwæðer, OE–ME ǽgðer, ME eigðer, (ME Orm.
eȝȝðer), ME æiðer, aieþer, ME eiðer, eiþer, ME aiþer, aither, ayther,
( ayder, ME eyder), ME–15 ether, (? ME–15 aþer, ather, athir, ME
euther, ewther), ME–16 eyther, eythir, (ME eithar), 15 eather, ME–
What's interesting is that some of these spellings seems to suggest that the modern pronunciation "should" be "AY-ther" (IPA /ˈeɪðər/) which of course is not correct: nobody actually says it like that.
I don't actually think we need to compare to German and Dutch, since so many earlier forms in English are attested, but in any case the cognate words are German jeder (IPA /ˈjeːdɐ/, roughly "YAY-duh"), and Dutch ieder (IPA /iːdər/, roughly "EE-der"). In fact, both of these would correspond to the English pronunciation "EE-ther" (IPA /ˈiːðər/) rather than "EYE-ther" (IPA /ˈaɪðər/). I think some people may be misled by their knowledge of the ei digraph used in modern German to represent /aɪ/, but in fact this is completely irrelevant with regard to the early Germanic etymology of the word, since the German word cognate to either is not spelled with ei.
The history of the word in English, as varied as it is, also seems to me to point more toward /ˈiːðər/ than /ˈaɪðər/. For example, there's the attested Middle English spelling "eather," and also a handful of other fairly well-established words where "ei" represents /iː/ such as "key," "ceiling," "seize" and "receive."
The OED seems to basically say the same, but notes that in practice both pronunciations have been recommended by the people who deal with such things:
The pronunciation /ˈaɪðə(r)/ , though not in accordance with the
analogies of standard English, is in London somewhat more prevalent in
educated speech than /ˈiːðə(r)/ . The orthoepists of 17th cent. seem
to give /ˈɛːðər/ , /ˈeːðər/ ; Jones 1701 has /ˈeːðər/ and /ˈaɪðər/ ,
Buchanan (1766) has /ˈaɪðə(r)/ without alternative (see Ellis, Early
Eng. Pron. ix, x.). Walker (1791) says that /ˈiːðə(r)/ and /ˈaɪðə(r)/
are both very common, but gives the preference to the former on the
ground of analogy and the authority of Garrick. Smart (1849) says that
‘there is little in point of good usage to choose’ between the two
pronunciations, though in the body of his dictionary he, like earlier
orthoepists, gives /ˈiːðə(r)/ without alternative.
However, there are a few other English words where "ei" represents /aɪ/. It occurs:
- in eye (if we consider "ey" a spelling variant of "ei")
- before "gh" in some words: height, sleight
- in words with Greek roots, which I don't believe were ever frequent enough to have had a significant influence on the pronunciation of the common word "either"
- in words that were recently taken from German, as mentioned earlier
One possibility I see for explaining /ˈaɪðər/ is that the word simply had an irregular phonetic development, possibly due to sound changes that only occurred in one dialect.
I suppose it is at least possible that the German digraph influenced English pronunciation along the lines laid out in @GEdgar's comment:
What I heard (another "hearsay" answer, so I just make a comment) was
that Prince Albert, being of German origin, used the pronunciation
suggested by German, and then it subsequently became popular among the
It seems unlikely to me, but worth mentioning.
It might also be valuable to compare this word's pronunciation with that of "aye."