🔼The name Abagtha in the Bible
Abagtha is one of seven court officials of king Ahasuerus, the one who married Hadassah or Esther (Esther 1:10). These seven are usually described as being eunuchs, but the word used is סריס (saris), literally meaning "of the king's head". HAW Theological Wordbook of the Old Testament reports that "the meaning eunuch arose with the practice of castrating men in key positions in the various nations of the ancient near east". Thus, whether Abagtha was a eunuch is not at all sure and certainly not implied by the text.
Abagtha is mentioned in one breath with his colleague בגתא (Bigtha), whose name is strikingly similar.
🔼Etymology of the name Abagtha
Neither NOBSE Study Bible Name List nor BDB Theological Dictionary try an interpretation but Alfred Jones (Dictionary of Old Testament Proper Names) believes this name comes from the Sanskrit verb bagadata, meaning given by fortune, which in turn comes from the noun baga, meaning either fortune or the sun. Jones curiously reads Fortune for Abagtha but Given By Fortune for Bigtha.
Although the Hebrew scribes often took considerable freedom in transliterating foreign names, and sometimes used this freedom to force a meaning into that name, the name Abagtha doesn't seem to mean anything in Hebrew. The first part may have reminded of the word אב (ab), meaning father. And the second part may look like it has to do with גת (gat), meaning winepress or wine vat, but these similarities are probably coincidences and have no obvious meaning.
If the etymology Jones offers is correct, both names mean Given By Fortune.