Greek had the /h/ phoneme only at the beginning of a word, and it was marked with a diacritic (rough breathing sign) rather than with a letter. Koine Greek lost the /h/ phoneme and early manuscripts (such as Codex Sinaiticus and Codex Vaticanus) didn't mark rough breathing, or had it added by a later scribe (according to their respective Wikipedia pages), and so they don't offer evidence on how it was pronounced. Latin, which did have represent /h/ with a letter, would translate the rough breathing as "h".
Since readers unschooled in Hebrew had to guess for themselves whether the Greek word began with a rough breathing mark, both Ierousalem and Hierousalem, Ieremias and Hieremias existed in Greek.
Alexander Sperber wrote an article with a list of Greek transliterations (Septuagint and Origen) and Latin transliterations (Jerome) of Hebrew words. Since the Greek words don't mark breathing, only the Latin is helpful here. Jerome uses i/j for this letter; he never uses "hi" to translate words beginning with Hebrew yod (his full list is on p. 124). Jerome, who learned Hebrew, certainly would have had no doubt as to the correct pronunciation.
Apparently, the rough breathing interpretation won out, because that was what was borrowed into Latin. It certainly helped that hieros meant "sacred" in Greek (that point made here), which would have been meaningful to Christians who considered Jerusalem a holy city, but also could have been a simple hypercorrection. Notably, names such as Iesous are never (to my knowledge, at least) spelled Hiesous, which might support the theory of a hieros misunderstanding, but if names such as Hiakobus are indeed attested, I can only explain this as having been made by analogy to words such as Hierousalem.
So the "hi" spelling in Latin Hierusalem, Hieremias, etc. would have necessarily come through Greek, and not directly from Hebrew.