Many local non-Jews refer to the Wall as the Wailing Wall based on the fact that Jews would cry when praying there. The name came from El-Mabka, or “the Place of Weeping,” the traditional Arabic term for the wall. After the commencement of the British Mandate, “Wailing Wall” became the standard English term.
According to Jewish Virtual Library. “Only after the Six-Day War in 1967 did it become de rigueur in Jewish circles to say “Western Wall”— a reflection of the feeling, first expressed by official Israeli usage and then spreading to the Diaspora, that, with the reunification of Jerusalem under Israeli sovereignty, there was no longer anything to wail about. Henceforward, the wall should be a place of celebration.”