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.