Here's an idea I've been mulling over. I'm 100% onboard with Noah as a real historical person whose family spread from Mount Ararat. So, looking at this purely from a Christian perspective, if I'm Noah, when my family goes out into the land there are going to be temples to Yahweh everywhere. There aren't going to be temples to any other gods, just to Yahweh.
According to the Jews, Noah was still alive in the time of Abraham; and according to Sumerian mythology he was granted eternal life.
As the eldest man of his family in a hard patriarchy, he should have had influence up to the time of Abraham.
According to Wikipedia, Abraham's great-grandfather Serug was the first to turn to the worhip of other gods - so every patriarch between Noah and Serug should have been filling Mesopotamia with temples to Yahweh, right?
Then why did they rebuild the cities of the Annunaki, and why were the Annunaki worshipped there? No one else could be responsible. Ur was rebuilt in the time of Serug, so he "might have" built that one; and introduced the worship of Nana; but I don't think he did any of the others.
But here's the real kicker - on this map we have Ur of the Chaldees (by the Persian Gulf), where Abraham, Terah, Nahor and Serug were from -
Take note of the closest city to Ur - Eridu, only ten miles or less away. Eridu, which the Sumerians considered the first city built - ever. Wikipedia says it is "among the oldest ruins in the region." There appears to be archaeological evidence for the Sumerian's claim - I don't know what the other old ruins are though.
In Sumerian mythology, Eridu was originally built, in pre-flood times, by Enki Ea - the Annunaki who created mankind and is an awful lot like Yahweh. Even his name looks like it could just plain be pronounced "Yah."
Now if Eridu is the oldest city, doesn't that imply that it was the first city Noah and his sons built after the flood? And they wouldn't have only rebuilt Ea's city first, but his temple as well -
The main temple to Enki was called E-abzu, meaning "abzu temple" (also E-en-gur-a, meaning "house of the subterranean waters"), a ziggurat temple surrounded by Euphratean marshlands near the ancient Persian Gulf coastline at Eridu. It was the first temple known to have been built in Southern Iraq. Four separate excavations at the site of Eridu have demonstrated the existence of a shrine dating back to the earliest Ubaid period, more than 6,500 years ago. Over the following 4,500 years, the temple was expanded 18 times, until it was abandoned during the Persian period
Link
In Sumerian flood myths, it is Enki Ea who instructs Noah (under the names Ziusudra or Atrahasis, but clearly the same guy) to build an ark and ride out the flood.
Ziusudra (Noah) is on the Sumerian King's List as the last King of Sumer before the flood.
Unless there was an earlier temple built in Northern Iraq that I don't know about, than it seems like Noah and his sons did make building a temple to Yahweh their first priority, and Yah is the same as Ea. Then they rebuilt the cities of the other Annunaki, because they were back on good terms again after the flood.
If that was fun, take a look at where Enlil Nunamnir's Nippur is on the map - between the Tigris and Euphrates. Same as Eden.