I think the situation as to pronunciation as it exists today has been pretty well covered by your other answers.
But I wonder if you might have found information about this from a source that was thinking more in terms of long term history. Withycombe's
The Oxford Dictionary of English Christian Names gives
Meriel and Miriel as common alternative spellings of
Muriel in medieval times, and says that the modern English surnames Merrell and
Merrill sometimes derive from an ancestress named
Muriel. This implies to me that back in medieval England the pronunciation of the first syllable of
Muriel rhymed more with the modern word "fur" than it does with "pure". In other words, original
Muriel didn't have that y-consonant sound in the first syllable that English speakers use today. Perhaps that somehow got into the name when it was revived in the 19th century.
And if back in medieval times some there were dialects where "ur" turned to "air", that would explain how the
Merrill is related to
Muriel.
Finally, in the dialect of most of the United States,
Meriel and
Mariel would today be pronounced about the same, just as most Americans west of New Jersey pronounce
Mary, merry, and marry the same. So perhaps the idea that
Mariel is ultimately a variation of
Muriel comes from some American linking them through the medieval spelling
Meriel? Just a thought. :)