We need to distinguish a mediaeval Latin pronunciation from an English one. The English pronunciation is what would be used in talking about him in a English context, and is what one might expect for "Boëthius" [boʊˈiːθiəs] ("bowEEthius") The alternative spelling "Boëtius" yields [boʊˈi:ʃəs] ("bowEEshus") or similar.
However, if one wants to know how he and his contemporaries pronounced his name, it was probably [boˈeːtsiʊs] ("boAYtsioos")
Explanation: The name is of Greek origin and would contain a theta in Greek. If one were to pronounce this in the Classical way, that would be [bo'e:t(h)iʊs]. By late Latin times, however, "th" was pronounced exactly like "t" (which is why "Boetius" is a mere spelling variant of the "correct" "Boethius"), and moreover the sequence [ti]+vowel when unaccented came to be pronounced [tsi]+vowel. Thus, [boˈeːtsiʊs] for Boet(h)ius.
This assibilated "t" would further develop, often to [s] in many areas speaking a neo-Latin dialect, and in particular France. And, after the Norman Conquest, French conventions for pronouncing Latin were imported into England. Thus, Latin natio went from Classical [natio:] to late [natsio] to French nation (with "t" pronounced [s]). The last step is to further assimilate [sj] to [ʃ] as in English "nation".
In summary, [th] -> [t] and [ti] -> [tsi] in late Latin; [ts] -> [s] in French; and [sj] -> [ʃ] in English.
Oh yes, forms like "Boece", "Stace", and "Lucrece" are French forms, imported into English.