There's a strong argument to say that there isn't a 'correct' way of pronouncing this word, and that both of the common variations are acceptable, but only one of them has found its way into Merriam-Webster's dictionary, so that's the one that we're going with, and we apologize to those people who are going to be upset by it. We figure you can't argue with the dictionary, though, so don't blame us if you're not happy with the result!
Whilst you've probably heard both "eye-thur" and "eee-thur" every day of your life for as long as you can remember, according to the dictionary, it's "eee-thur" that wins the debate. We don't think that's going to stop people debating, and we think people should be free to use either. Or that should be, free to use "either" however they want.