The origin of the name isn't certain. We do know it predates the Arab conquest in the seventh century, so the origin isn't in Arabic.

Your suggestion that the name could come from Uruk is indeed one possibility. The origin of that name comes from the Sumerian word for city or village.

It's also possible the name has an origin with the middle Persian word eragh, which meant "lowlands." As for the significance in this theory, here's an excerpt from Paradise and Paradigm: Key Symbols in Persian Christianity and the Baha'i Faith by Christopher Buck:

According to Wilhelm Eilers, the name Iraq is actually a Persian word (eragh), meaning "lowlands." ... As Eilers observes, "For the Sasanians, too, the lowlands of Iraq constituted the heart of their dominions." This shows that Iraq was not simply part of the Persian empire—it was the heart of Persia.

So we have a couple possible origins here, both with some clear significance to the region.

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