🔼The name Madai in the Bible
The name Madai is applied to one man and one country, namely Media, which is known mostly for its contribution to the great Medo-Persian empire.
Madai the man was a son of Japheth, son of Noah (Genesis 10:2). We don't hear from Madai after his introduction in Genesis 10:2 (apart from a similar reference in the parallel text of 1 Chronicles 1:5), but since the people's list of Genesis 10 appears to explain the rise of the various nations known in Biblical times (see Genesis 10:5, 10:20 and 10:31-32), Madai could probably be understood to be the progenitor of the country by this name: מדי which denotes both the country Media, and the people who lived in it; the Medes.
Media doesn't receive a lot of screen time in the Bible, and this is because it was subjected to Assyrian rule until less than three decades before the fall of Jerusalem to the Babylonians in 586 BC. When the Assyrian empire Shalmaneser V deported the people of Samaria, he took them to Assyria and the cities of the Medes (2 Kings 17:6 and 18:11).
The Persian Cyrus, or Darius, the Great (whose father was king Astyages of Media, against whom he rebelled; see Daniel 9:1) annexed Media to the Persian empire in 550 BC and conquered the Babylonian empire in 539 BC. This event appears to have been foretold long before by the prophets Isaiah (13:17 and 21:2) and Jeremiah (25:25, 51:11 and 51:28), and it happened one year before Cyrus/Darius issued the decree that allowed the Jews to return to Canaan (Ezra 1:1-4).
About a third of the references to the name Madai (Media or Medes) in the Bible occurs in the Book of Esther, which plays around the year 480 BC, when the Medo-Persian empire was at its apex, and about 44% of the world's population lived within its borders (says Ehsan Yarshater in The Cambridge History of Iran). The Medo-Persian empire fell to Alexander the Great in 330 BC.
Media consisted of six tribes, among which the Magi to whom the famous Zoroaster or Zarathustra was born, and it's likely that the Magi that went looking for the new born Messiah (Matthew 2:1) were not simply travelling illusionists as is sometimes supposed, but Median Magi. That by the first century AD the Medes were still recognized as an autonomous ethnic group is demonstrated by Luke's report of Medes (spelled Μηδοι, plural of Μηδος, Medos, a Mede) being among the various foreign visitors to Jerusalem who heard the disciples speak in their native languages (Acts 2:9).
🔼Etymology and meaning of the name Madai
The etymology of the name Madai is unclear, and some scholars have proposed that it was originally an epithet having to do with medium or middle, which described the land geographically (perhaps comparable to the name of the Netherlands, which originally described the lower part of the Habsburg empire). In Hebrew, however, the name Madai appears to be associable to the verb מדד (madad) meaning to measure:
Add to the word מד (mad) the possessive yod and the word מדי, meaning "my measure", appears. Alfred Jones (Dictionary of Old Testament Proper Names) derives our name indeed from the verb מדד (madad) but assumes that the final yod is a remnant of the appellative יה (Yah) = יהו (Yahu) = יו (Yu), which in turn are abbreviated forms of the Tetragrammaton יהוה, YHWH, or Yahweh. Hence Jones reads for a meaning of the name Madai: Extended Of The Lord.
Another way of looking at this name is through the word די (day), meaning sufficient or enough:
This word is often combined with certain prepositions, to make more specific forms of sufficiency. One of those combinations is with the letter mem: מדי = די + מ, meaning "out of the abundance of", or "as often as" (according to BDB Theological Dictionary).
Hence, apart from Jones' solution, the name Madai may also mean My Measure(ment) and Out Of Abundance.