I answered a similar question here.

In the case of Columbus, he would have been variously known as (Italian) Cristoforo, (Spanish) Cristobal and (Portugese) Cristovão, with variants. Similarly e.g. Charles V knew something like a half-dozen languages (Spanish, French, German, Flemish, Latin...) that would variously have rendered his name Karl, Carlos or Carolus. There are no obvious choices here.

But the choice is ultimately a bit arbitrary. To use my own area of interest as an example, in Middle Iranian languages I usually render the names approximately natively - Ardaxšir rather than Artaxares, Aršak rather than Arsaces, Khusrau r.t. Chosroes, Kavad r.t. Cabades, etc, and almost all Iranologists do the same, with some spelling variants (Husrau, Khosrow...). On the other hand, basically nobody renders Achaemenid names in Old Persian; they're too obscure and the pronunciation is too uncertain. Xerxes would become Xšayarša (rules over men, note that X is used for the Persian Kh sound in the latter, as in Khomeini) but was probably pronounced something like Xšyərša (IIRC there are Elamite renderings of the name similar to this). Dareios would become Daraya-Wahuš (doer of good), but again, it's likely (but far from certain!) to have been pronounced along the lines of Daryəuš. And even though I usually go with the Greek forms of names I tend to use Cyrus rather than Kyros, just because the latter is so much less familiar and pronounced so differently.

There are no hard-and-fast rules, it's ultimately a matter of who you're communicating with.

In an East Asian context, /u/Parallelpain discussed in a previous thread the problems of transliterating historical names in Japanese, where characters can have multiple pronunciations.

In my own area, of course, there is a bit of a convention whereby certain individuals are arbitrarily known by a certain transliteration (Sun Yat-Sen and Chiang Kai-Shek spring to mind), as well as (in Hong Kong studies) most people who used a certain transliteration in their own lifetimes (eg. Robert Hotung, Ho Kom Tong). However, there's a tendency to use Mandarin transliterations for even for speakers of Cantonese, Hakka, Hokkien, Shanghainese and other 'dialects' (they're really not), even, in some cases, among Hong Kong specialists, in no small part due to the lack of unified standard Romanisation methods for any of them. Wang Jingwei, the infamous defector to the Japanese in WW2, was Cantonese and so usually would have said his name (in private at least) as Wong Jing-waih; Zheng Chenggong (a.k.a. Koxinga), the Ming loyalist who conquered Taiwan, would have been known as Ti Seng-kong; Hong Xiuquan, ruler of the Taiping Heavenly Kingdom, would have been Fung Siu-chhion in his native Hakka (and more than a few contemporary Western documents use a variation on Hung Sawchuen, derived from the Cantonese). The problem of native language here is a significant one, and not one I often (practically ever) see addressed in the literature, but part of it simply comes down to Mandarin being the dominant (public) Chinese language and therefore the first and likely only Chinese variety most specialists will learn.