It's fairly certain that the Roman pronunciation of Niobe was not the same as the pronunciation used by modern English speakers (aside from the consonants, which I won't discuss further in this answer).
Please don't misinterpret this to mean that the English pronunciation is "wrong"—it's perfectly correct, in English. It's just different from the Latin pronunciation, just as the acceptable English pronunciations of the word status are different from the pronunciation we think was used for this word by the Romans.
In Latin, the forms Nĭŏba and Nĭŏbē were both used according to Lewis and Short as accessed through Perseus.
A phonemic or broad phonetic transcription for the pronunciation of Nĭŏbē would be /ˈniobeː/. The stress would have been on the first syllable, due to the normal Latin rules for assigning stress. Latin was not pronounced with the exact same sounds as English, so any attempt to write this out using English sounds can only be approximate. That is why I am using the International Phonetic Alphabet (IPA). If you are not familiar with it, here is a chart showing some of the vowel sounds in IPA and their relative positions, with audio files representing the general sound that each symbol is used to represent: IPA Chart With Sounds. You can see that the IPA is more systematic and includes more vowels than the English sound system, so even though it is also an approximation, it allows us to be a bit more precise.
Phonemic vs. phonetic transcription
I said earlier that the transcription /ˈniobeː/ I gave was "phonemic." This means that it only has the essential details needed to distinguish the sounds from other sounds used in Latin. It does not describe the details that are not contrastive.
In Classical Latin, some of these non-contrastive details were related to vowel "quality," or the position of vowels in the mouth. My understanding is that Latin basically started out with only 5 distinct vowel qualities (the familiar /a e i o u/); the main distinction between short and long vowels was vowel "quantity," whether a vowel was phonetically long or short. But over the course of Classical Latin, most pairs of vowels with distinct quantities began to evolve to also have distinct qualities, and phonetic length distinctions were eventually lost entirely in Vulgar Latin, the ancestor of the Romance languages.
I'm going to try to explain the quality of some vowels in Classical Latin; to do this, I'll use two types of IPA transcriptions. Broad transcriptions, such as /ˈniobeː/, are enclosed by square slashes, and don't include any information about non-phonemic details of vowel quality. For narrow transcriptions that do describe vowel quality in detail, I'll use square brackets.
Pronunciation of ŏ: [ɔ]
The Latin short ŏ phoneme (IPA /o/) seems to be fairly widely agreed to have been pronounced with the phonetic vowel quality [ɔ], both in the Classical period and in Vulgar Latin. If you want approximations for this sound in English: for a British English speaker, it's similar to the vowel used in the word lot. For an American English speaker, it's similar to the vowel used in the word more, but with the "r" removed.
The pronunciation of the other two vowels is more complicated.
Pronunciation of ĭ: [i] > [ɪ] (> Vulgar Latin [e])
Originally, it seems likely that the Latin short ĭ phoneme (IPA /i/) was pronounced with the same vowel quality as long ī /iː/. The most likely phonetic value for short ĭ at this point would be IPA [i] (about the same as the English vowel in bee, but distinctively short). However, by the time Latin split into the Romance languages, this vowel is believed to have lowered to [e] (approximately the vowel in French clé, which often sounds to English speakers like our diphthong /eɪ/ as in "clay").
In between these two stages, it's likely the vowel passed through an intermediate quality transcribed [ɪ], which is approximately the same sound as in English "bit." There doesn't seem to be total consensus about whether short ĭ was pronounced [i] or [ɪ] in early Classical Latin.
In Vox Latina, W. Sidney Allen recommends that English speakers (specifically, speakers of "Received Pronunciation") use the vowel in pit for Latin short ĭ, and says that they "had much the same value" (p. 49). So, this is certainly acceptable, and would not qualify as a "butchering."
Personally, though, I find it awkward to try to use this vowel quality before another vowel or word-finally. In Received Pronunciation, /ɪ/ can occur in these positions, at least in unstressed syllables (in words such as "baryon" /ˈbærɪɒn/ and "happy" /ˈhæpɪ/) but in my accent of English, these words use /i/ instead. So I just use [i] in these contexts when pronouncing a word in Latin (but I still try to maintain a distinction in phonetic length).
Allen seems to suggest that using a more [i]-like quality before vowels was actually a feature of Roman pronunciation in the following passage:
The Latin short i [...] may well have had a closer quality (more like
that of the long ī) before vowels, to judge from the Romance
developments of Latin dies (Italian/Old French di, as chi/qui from
qui); this is indicated also by the fact that i is scarcely ever
written as e in this position (cf. p. 49), and is indeed often written
with the I longa (e.g.
prIusquam, dIes, pIus). There is a close
parallel to this situation in English, where the first vowel of e.g.
react is closer than that of recall, being more similar in quality to
the long ī; English speakers of Latin will therefore also
automatically make this adjustment. These closer pre-vocalic qualities
of e and i are probably due to the y-' glide' which automatically
follows them in these conditions—and which the English speaker will
automatically produce. (p. 51-52)
Furthermore, according to Andrea Calabrese's paper On the Evolution of the Short High Vowels of Latin into Romance, short ĭ in any environment was probably still [i] in early urban Classical Latin (around the time of Cicero) and quality distinctions between long and short versions of the same vowel only became established around the first century A.D. (p. 73). So I think [i] (a short version of the vowel sound in English bee) is defensible in general for short ĭ as an approximation of the early Classical Latin pronunciation.
Pronunciation of ē: [ɛː]? > [eː] (> Vulgar Latin [e])
Originally, it seems likely that long ē /eː/ was pronounced with the same vowel quality as short ĕ /e/. Calabrese argues that it's most likely that the phonetic value of long ē at this early stage was [ɛː] (about the same as the English vowel in "bed," but distinctively long).
But long ē eventually developed a distinct vowel quality from short ĕ; we have evidence that this had occurred already in Late Classical Latin (and possibly earlier). The quality of long ē was "raised" closer to an i-like sound, resulting in [eː]. You'll notice that this is the same vowel quality as the one I described earlier for short ĭ; in fact, Latin short ĭ and long ē generally evolved to the same vowel in Romance languages.
Here's Allen's description of this sound:
Long ē and ō present rather greater difficulty for R.P.* speakers,
since this dialect contains nothing really similar; the nearest are
the sounds of e.g. bait and boat-but these are very distinctly
diphthongs, [ei] and [ou] respectively, which the Latin vowels were
not. (p. 50)
Putting it all together
So the Roman pronunciation would likely sound something like "NEE-ob-ay" to a British English speaker, or "NEE-o(r)-bay" to an American English speaker (where "o(r)" represents the vowel sound in the word more, but without the "r"). Unlike in English, the last vowel would be a single, pure long vowel sound rather than a gliding diphthong.