Is there a rule in British English about how to pronounce "either"?

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).

I think the variation in British usage is almost all between users, not between examples of use. It's mostly a regional and class distinction: I grew up saying /ˈaɪðə/, and we looked down on people who said /ˈiːðə/.