Why do people pronounce "Naomi" as "Niomi"?

I think this is a mild hyperforeignism that comes from an attempt to pronounce “Naomi” more like the original Hebrew: nah-oh-mee [na.o.mi]. The [ao] sequence is uncommon in English—and because there are two separately stressed syllables in this case, they cannot merge into ow [aʊ]. Thus an epenthetic /y/ [j] sound appears, giving nah-yo-mee [na.joʊ.mi]. This is just like someone saying “drawring” instead of “drawing”: the transition between a certain pair of vowels is uncommon, so a consonant appears to simplify pronunciation.

The reason you hear it as nye-oh-mee then becomes clear: an [a] sound followed by a [j] sound approximates the regular English “long I”, which is in fact a diphthong, and not a single vowel. As a similar example, consider the word diode, which could be rendered as dye-ode [daɪ.oʊd], but also as dah-yode [da.joʊd].

So it’s a matter of principle: call your daughter by the name (pronunciation) she was given, but also acknowledge that neither pronunciation is really any more “right” than the other. If someone wants to name their daughter Naomi (or son, whatever, it’s a free country) but pronounce it “squeemdge”, then that’s their problem, and they’ll just have to correct people. Often.

The name Naomi is commonly pronounced nay-OH-mee in the United States. But as with all Hebrew/Biblical names, one will find several variations in pronunciation since they are not originally English names. Apparently, nigh-OH-mee is also accepted, according to Behind The Name. In my view, this is less common, but I do not have an answer for why some people would choose to pronounce it this way. I grew up in church in Nigeria, and we always called it nah-OH-mee in Sunday School. If you do some research, you will find that this is closer to the actual Hebrew pronunciation.

Another popular Hebrew/Biblical name is Aaron. In America, these days, it is pronounced almost like the name Erin. But some swear by AY- for the first syllable. Apparently, AH- is closer to the original Hebrew.

As with all names, I would go by the commonly accepted pronunciation or stick with how the owner of the name (or their parents) addresses them[selves].