Names present an interesting dilemma for translators. Do you:
- Translate the literal meaning? (e.g. Cephas -> Petros)
- Transliterate the pronunciation? (e.g. Yeshua)
- If the person goes by different names in different languages--common in the Roman Empire--do you use their commonly known name in another language? (e.g. Levi -> Matthew)
- Use a variant of a name that has been translated twice? (or more) (e.g. Messiah vs. Christ)
- Or some hybrid among these options?
Option 4 above is particularly noteworthy in a comparison between the KJV & the DRB. Most of the Old Testament was written in Hebrew, and at least the vast majority of the New Testament texts were originally written in Greek, but the DRB relied heavily upon the Latin Vulgate, which made it, in many places, a translation of a translation.
The Old Testament names in the KJV exhibit an interesting provenance as well. They were recorded in the Tanakh in Hebrew, translated into Greek when quoted in 1st-century Christian texts, and then translated from Greek into English by the KJV translators. This is why in English the name of a prophet (e.g. Isaiah, Esaias) is sometimes different in the Old Testament versus the New. One text was translated from Hebrew to English, another was translated from Hebrew to Greek to English, and influenced by other languages as well.
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Common names are more consistent than uncommon names
"David" is an example of a name that has been in wide circulation for 3,000 (or more) years. Most languages with Judeo-Christian influence have a common name for "David"; translators do not need to transliterate or develop their own translation, they can just use the existing, common name. As a result, the name David has a rather consistent spelling in English Bibles.
Other names are not widely used (when was the last time you met a person named Habakkuk?) and so a language may not have an existing, well-known version of it. There is not a single set of translation policies to dictate which of the options above a translator should use. If there's a way this name is used in speech (i.e. people have been preaching in a language before the book was actually translated into the language), the translators are likely to just spell phonetically the way people say the name. But even transliterations are approximations and will vary from place to place.
Even then, some of it just boils down to taste. When speaking Spanish and referring to my American friend Thomas, do I call him "Thomas" or "Tomás"? Honestly, I've done both.
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Language development
Additionally, as already noted by Michael16, languages change over time, and spelling, pronunciation, and even meaning of words will be different from generation to generation (though the variation is much more evident when comparing across several centuries). Wycliffe's translations of Biblical texts into Middle English, in the 14th century, make very difficult reading for speakers of modern English.
Both the DRB & KJV have been through multiple revisions since their original publication in the 16th & 17th centuries, respectively. A Bible translation written into 23rd century English a few hundred years from now, would doubtless include major variations in wording, spelling, and pronunciation from a modern English Bible today.
In some cases, the original pronunciation of a word has been lost (this is particularly common in ancient languages, like Hebrew, that were written without vocalization), and so one scholar's opinion on how to pronounce & transliterate a name will differ from another's.
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Conclusion
Preference for KJV names, pronunciations, and even verbiage is common, because the KJV has had such an enormous impact on the English language. Many words & phrases in the KJV have entered common speech, and are still used even though they would be seldom employed otherwise.
Even when other translators differ in opinion with the KJV translators on the best way to render a name from one language into another (and again, there's no clear, hard and fast rule here), the way the names were rendered in an extraordinarily influential text, like the KJV, is likely to remain in circulation for many years to come.