The story
From Chinese, though ultimate source language unascertained. Apparently 少底, literally "less base", from 少 (“less”) + 底 (“base; bottom”) in Cantonese pronunciation, referring to the condensed milk, used for sweetening, that sits at the bottom of the cup. In Cantonese this would be pronounced siu² dai², but the compound has not been traced in Cantonese.
Etymology from Wiktionary, CC BY-SA 4.0.