At least 3 men named Elam appeared in the Old Testament. More than likely, the region called Elam was named after the first-born son of Shem.
Both Genesis 10:22 and 1 Chronicles 1:17 identified
Elam as a son of Shem (Semite).
Noah > Shem > Elam
Brothers: Asshur, Arphaxad, Lud and Aram
One of 11 sons (or descendants) of Shashak, from the tribe of Benjamin, was Elam as stated in 1 Chronicles 8:22-25.
Shaharaim > Elpaal > Beriah > Shashak > Elam
Brothers: Ishpan, Eber, Eliel, Abdon, Zicri, Hanan, Hananiah, Anthothijah, Iphdeiah and Penuel
During the time of king David, 1 Chronicles 26:1-3 stated Elam as one of 7 brothers of Meshelemiah
Asaph > Kore > Meshelemiah > Elam
Brothers: Zechariah, Jediael, Zebadiah, Jathniel, Jehohanan & Eliehoenai
For several reasons, the second and third men apparently were the ones mentioned in this passage. Ezra 2:1-7,31:
(the same 2 men are mentioned in Nehemiah 7:6-12,34)
1. Now these are the people of the province who came up from the captivity of the exiles, whom Nebuchadnezzar king of Babylon had taken captive to Babylon (they returned to Jerusalem and Judah, each to his own town,
2. in company with Zerubbabel, Jeshua, Nehemiah, Seraiah, Reelaiah, Mordecai, Bilshan, Mispar, Bigvai, Rehum and Baanah): The list of the men of the people of Israel:
...
7. of Elam 1,254
...
31. of the other Elam 1,254
Two more descendants of Elam surfaced in the book of Ezra.
Ezra 8:7:
1. These are the family heads and those registered with them who came up with me from Babylon during the reign of King Artaxerxes:
...
7. of the descendants of Elam, Jeshaiah son of Athaliah, and with him 70 men;
Athaliah > Jeshaiah
In Nehemiah 10:14-27, a man named Elam was described as a "leader of the people".
Additionally, Nehemiah 12:42 mentioned a Levite who returned with Zerubbabel from captivity in Babylon:
40. The two choirs that gave thanks then took their places in the house of God; so did I, together with half the officials,
41. as well as the priests--Eliakim, Maaseiah, Miniamin, Micaiah, Elioenai, Zechariah and Hananiah with their trumpets--
42. and also Maaseiah, Shemaiah, Eleazar, Uzzi, Jehohanan, Malkijah, Elam and Ezer. The choirs sang under the direction of Jezrahiah.
It's unclear if the men stated in Nehemiah 10 & 12 were the same from Ezra or if they were different people altogether.