The story of the name
They ask you about him
"And they ask you about Dhul-Qarnayn" (Surah al-Kahf, 18:83). The question came to the Prophet ﷺ directly. What followed was one story, told in full, found only in Surat al-Kahf. The Qur'an gives him no name. It gives him no nation. It gives him no era. What it gives instead is his character.
Established in the land
"Indeed, We established him in the land, and gave him a way to everything" (Surah al-Kahf, 18:84). His power came from Allah. The Qur'an says so plainly, before it tells us anything else about him.
Journey to the west
He traveled west, until he reached "the place where the sun appeared to set in a spring of dark water" (Surah al-Kahf, 18:86). Commentators read this as the sun's appearance from where he stood. Not a literal claim about where the sun goes. There he found a people, and Allah gave him a choice: punish wrongdoing, or show mercy. His answer held both. "As for the one who does wrong, we shall punish him... but as for the one who believes and acts righteously, he shall have the best reward" (Surah al-Kahf, 18:87–88). Justice, and mercy, in the same breath.
Journey to the east
He then traveled east, "until he reached the rising place of the sun" (Surah al-Kahf, 18:90). He found a people with little shelter from it. The Qur'an gives no more geography here. The attention stays on him, not on the place.
The barrier against Gog and Magog
His most famous act came between two mountains. A people there were suffering. "Indeed, Gog and Magog are causing corruption in the land" (Surah al-Kahf, 18:94). They offered him payment to build a barrier against them. He refused it. "That in which my Lord has established me is better" (Surah al-Kahf, 18:95). He asked only for their labor. He built the barrier from iron, then poured molten copper over it (Surah al-Kahf, 18:96). It held. Gog and Magog could not cross it, and could not breach it.
A mercy, and a temporary one
He did not take credit for what he built. "This is a mercy from my Lord. But when the promise of my Lord comes, He will level it to the ground" (Surah al-Kahf, 18:98). Even his greatest work, he said, would not last forever. Only Allah's decree does.
To carry this name
To carry this name is to hold power the way Dhul-Qarnayn held it — refusing payment for good you're able to do, and giving the credit for every success to Allah, not to yourself.