There are two common pronunciations of "either": British /ˈaɪðər/ and American /ˈiːðər/. If Americans are more or less consistent in this regard, then the Brits seem to be freely using both. In fact, from what I can tell, "either this or that" is more often in the first form, whereas "me either" is in the second. But I may be wrong. Is there any kind of an informal rule in the modern British English with regard to this?
Usage of /i:/ ("EE-thur") and /ai/ ("EYE-thur") in Great Britain and in Canada seems to be mixed. In the United States, the predominant usage has always been /i:/. However, there's also a long history of /ai/ occurring among a few Americans, including Benjamin Franklin and James Fenimore Cooper in earlier times, and Franklin Roosevelt, Ronald Reagan and Barrack Obama more recently. For many years, /ai/ was associated in the United States with British usage and, by extension, with certain elites who tried to imitate British usage. It seems to have become more widespread in recent decades, however. While some Americans have adopted /ai/, perhaps because they feel that it sounds sophisticated, other Americans regard it as pretentious.
Spelling, by the way, has nothing to do with the difference. In English, the spelling ei usually represents the "long a" pronunciation (IPA /ei/), as in eight, feign, or rein. In such words, it is derived from the Middle English /ai/ diphthong, which normally developed into the "long a" sound. In a smaller set of words, such as receive, ei represents the "long e" sound /i:/. It's rare for ei to represent the "long i" sound /ai/ in words that have been in English more than two or three hundred years; most words spelled with ei and pronounced with "long i" are recent borrowings, such as Poltergeist (from German), or other words that only recently developed a standard spelling, such as heist (originally a variant of hoist).
If you listen to Ira Gershwin's lyrics, you realize that this argument has been going on for quite some time in America as well.
You say "either" and I say "either"
You say "neither" I say "neither"
"Either" "either", "neither" "neither"
Let's call the whole thing off
The song points up for comic effect the differences between two lovers, who come from different social strata. Although few Americans would say "potahto" or "tomahto" these days, there still remains a divide in the pronunciation of either, sometimes within the same speaker. Many Americans, even ones who use what you call the British pronunciation most of the time, will say "eether" in constructions like "an 'either/or' proposition".
I find myself using the "ee" pronunciation most of the time, but switching to the other for emphasis: "Either you help me with this project right now or I won't have time to help you with yours."
There are other words which are similarly split with regard to pronunciation to the extent that there is no "correct" way to say them. Envelope comes to mind. Many Americans pronounce the first syllable of that word to rhyme with on. I don't think you can make any hard and fast rules; it just depends on who you are and what you grew up with.
According to The Complete Atlas Of The British Isles, "eether" is more common in the north of England and "eye-ther" in the south of England, with the Midlands and London using both. The atlas also mentioned a THIRD way of pronouncing the word, "ay-ther", in parts of the north of England, which I suppose is now extinct.
I'm in Canada. Typically I pronounce it "ee-ther." I do notice a lot of American television dialogue uses "eye-ther," and American English has influenced Canadian English pronunciations since the radio and television age.
I say "me neither" (as in 'neether'). I never say "me either", although it doesn't really make less sense than "me neither", when properly it should be "nor I."
I actually believe that there are some Germanic influences in the British English pronunciation of the word. In German, you would read
eider like /ˈaɪðər/. And in British English, you read
either the same way. So I believe that there are these Germanic influences, English being a West-Germanic language. But this is nothing certain, just a guess, as I've also been thinking about the differences between the two.