🔼The name Bigthan in the Bible
The names Bigthan (בגתן) and Bigthana (בגתנא) both occur only once in the Bible, and they denote the same person, namely one of the two eunuchs of king Ahasuerus who conspired against him (Esther 2:21, Esther 6:2). Mister Bigthan(a) and his buddy Teresh (who weren't necessarily eunuchs — see our article on the similar name Abagtha) where found out by Mordecai, queen Esther's uncle, and subsequently hanged on a tree (Esther 2:23).
🔼Etymology of the name Bigthan
The name Bigthan or Bigthana is Persian and doesn't mean anything in Hebrew. According the Alfred Jones (Dictionary of Old Testament Proper Names), this name comes from a Sanskrit word baga, meaning either the sun or fortune.
For a meaning of the name Bigthan, NOBSE Study Bible Name List reads Gift Of God and Jones' Dictionary of Old Testament Proper Names proposes Given Of Fortune. BDB Theological Dictionary confirms this name comes from Persian, translates it with Gift Of God, but adds a question mark to indicate that this interpretation is not entirely certain.