I was told several times that the pronunciation of either as "ee-ther" was American English whilst pronouncing it like "eye-ther" was British. Only: I do not know anybody from the UK who says "eye-ther" -- even in British films they use the assumed American version. Is "eye-ther" dying? Does only the Queen use it? Does it therefore sound strange when I use it?
There is a song about this: Let's Call The Whole Thing Off Lyrics by Fred Astaire
I love Fred Astaire, and he sang it, but the Gershwins wrote it.
What's odd is that I probably say "neye-ther" more often than "nee-ther" but I think I say both versions of either!
I do say "me neether" - but then, I say "neye-ther do I"... so... who knows?
Longman polled panel preference in Britain in1988: 88% /awðə/, 12% /i+ ðə/ and 1993 AmE poll: 16% / awðd/ vs 84% /i+ ðd/. Please note the 1st pronunciation is the "eye-ther" version and the 2nd "ee-ther" version.
Both forms of pronouncing the word "either" are correct in both varieties of English. So feel free to pronounce it as you feel like it. However, the ,let's say, most used form is ee-ther. I personally prefer eye-ther
I say eye-ther and n-eye-ther.