archaeology - What is the source that the Western Wall was built by the poor?

I understand that there is a general Jewish tradition that the Western Wall/Kotel was built by the poor (I believe under the command of King Solomon) (link). What is the source for the tradition (being built by the poor, but also when it was built)? All archeology has uncovered is that it was built after the year 20 CE (link).

From Aish:

2) Eternal Symbol

The Sages prophesied that even after the Temple's destruction, the Divine Presence would never leave the Western Wall, and that the Wall will never be destroyed. The Wall is endowed with everlasting sanctity, as the Talmud says: "And I will make your sanctuaries desolate" (Leviticus 26:31) – this means that the sanctuaries retain their sanctity even when they are desolate.

Jerusalem was destroyed and rebuilt nine times. And through it all, one symbol remained intact: the Western Wall.

5) Built with Love and Dedication

When the Temple was being built, the work was divided among different sectors of the population. The building of the Western Wall fell to the poor, and they worked hard to construct it, as they could not afford to hire laborers to do their work for them.

When the enemy destroyed the Temple, the angels descended from on high and – spreading their wings over the Wall – said: "This Wall, the work of the poor, shall never be destroyed." (from "Legends of the Land of Israel")

From NBC News regarding archeological finds (this archaeological data is available elsewhere, I just figured a news source would be considered reliable). I understand it doesn't seem to fit in line with Aish's article, but I'm bringing it here as it's the only physical evidence that relates to the subject of my question:

Newly found coins underneath Jerusalem's Western Wall could change the accepted belief about the construction of one of the world's most sacred sites two millennia ago, Israeli archaeologists said Wednesday....

The coins were found inside a ritual bath that predated construction of the renovated Temple Mount complex and which was filled in to support the new walls, Reich said....

They show that construction of the Western Wall had not even begun at the time of Herod's death. Instead, it was likely completed only generations later by one of his descendants.