Either - neither (pronunciation)
The words either and neither have two permitted pronunciations, even in the most formal speakers of British English. Some make the first syllable rhyme with 'eye' and 'sigh', IPA: /aɪ/. Others make it rhyme with 'be' and 'see' (/iː/). (You can hear both in the song "Let's Call the Whole Thing Off" by George and Ira Gershwin.)
A survey in 1988 showed that in British English, over 80% of speakers used the 'eye' pronunciation; in American English in 1993, over 80% used the '-ee-' pronunciation. If you want to sound British, say 'EYE-ther'; if you want to sound American, say 'EE-ther'. Or be like the founding editor of AWE, and vary the pronunciation to suit the phonetic context, and your current mood.
(See also either for a note on the meaning, and neither - nor for a note on grammar.)