So which pronunciation is standard for the [ʊ] sound? Rounded or unrounded?
Certainly there is some rounding, but because roundedness is not phonemic in this position, there is also considerable variation in how much of it actually occurs in any given word and speaker.
For example, you will find that it is generally somewhat more rounded in pull and full than it is in put and foot respectively. That’s because having an r or an l right next to it rounds it off a bit — which is why it is a bit more rounded in root and rook than it is foot or cook. Same with rookie versus cookie, where the first version is a bit more rounded than the second. And of course, a w helps: compare how wool is even more rounded than full, and also moreso that wood.
I believe English has no words with [ʊw], as that seems redundant. However, it can occur in phrases, especially in some dialects, where something like I knew it full-well may approach that.
However, it is still perceived as the very same phoneme in all those words and cases I’ve just listed above.
Correction — or not
I said that I thought English had no words with [ʊw] in them. And at the end of the day, I still believe that. However, I have discovered that grepping the OED yields the apparent existence-proof counterexample of Rauwiloid, which means:
A proprietary name for a hypotensive preparation containing a number of alkaloids extracted from Rauvolfia serpentina.
You also have compound words whose first element ends in [aʊ] (rather than [aw], as it is sometimes spelled) connecting to something that begins with [w], and which have in effect a “double w” in them, you expand the list to include such things as:
bow-wow, powwow, skeow-ways, wow-wow
Finally, if you consider the sound in words like no and micro to be
an [oʊ] diphthong rather than [ow], then you get all these, most of which were originally compounds of some sort:
froward, frowardly, frowardness, glow-worm, Holloway,
hollowwort, Howeitat, Khowar, meadow-wink, microwave, microweld,
Moldo-Wallachian, nowise, Oldowan, Parowax, powan,
shalloway, slow-worm, swallowwort, werowance,
yellow-wood, yeowoman.
For example, yeowoman theoretically yields /ˈjoʊwʊmən/, at least in North America. Still, there is a reasonably convincing argument to be made that that one is better written as simply /ˈjowʊmən/.
Slightly less uncommon is nowise, which is a compound of one word ending in a diphthong connected to another starting with a triphthong, so /ˈnoʊˌwaɪz/.
But I am still highly dubious of the existence of [ʊw], because I think it fuses into the semi-consonantal glide, [w]. After all, nowise and no eyes are homophonic, so I think this idea of [ʊw] is very hard to justify, and so I stand by my initial statement.
Even towel is usually pronounced with just one syllable, /taʊl/, thereby rhyming with cowl /kaʊl/. Even with folks who work very hard to put two syllables into that, with /ˈtaʊ.wəl/, I submit that you could write that /ˈtawːəl/ and avoid the whole controversy of whether a semi-vowel/semi-consonant/off-glide is really /ʊ/ or really /w/. However you write it, it seems like the same sound to me, such that bisyllabic towel just has a geminate [w]: /ˈtaw.wəl/.
Divine is not usually used as a name. Names like Devin (ranked 173 most common in the US), Devon (397), Davin (768), Devan (965) are much more common names than Divine, which doesn't even rank in the top 1000 or even appear in any of the name databases I looked in. Because people know that what they're trying to pronounce is a name, they're more likely to pronounce it like other names they've heard.
We interpret new information based on past information, assuming likelihoods based on past experience, in what is known as the availability heuristic. Because most people have never heard "divine" used as a name before, most people who see your name for the first time probably try to associate the spelling with names that they do know.
Your name is presented as a name, but (as you point out) "divine" is usually heard as an English word. Because people are viewing it as a name, they're not associating it with the English word. You don't meet a person named Ryder and consciously think of equestrianism or a person named Allan and consciously think of wrenches or a person named Jasmine and consciously think of rice or tea. You're viewing the names as names, and their alternate definitions as English words are processed subconsciously, if at all.
One possible, less likely, reason for the mispronunciation is that because "divine" is not a common name, people might actually think of its definition as a word and be hesitant to call you "divine" because of the various connotations the word carries with it. They could worry about how you might react if they pronounce it /dɪˈvaɪn/
if that turns out to be wrong. They could think: maybe she'll be offended by me giving a religious pronunciation of her name or maybe she'll think I'm giving a shallow compliment to someone I've just met.
Best Answer
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.