점심 comes from 點心, which is commonly known to westerners as dimsum. Dimsum, by the way, is not lunch for Chinese people. It’s more like brunch and it is mostly a Cantonese thing.
점심 comes by the way of sound changes: 뎜→점 and 슴→심. In the same manner, you get the common family name 김 from 금 (金)!
In Japan it's written 点心 ("tenshin", the first character is always written this way) and means snacks eaten with tea, but most people know it as "yumcha" (飲茶, literally drink tea) which is supposedly the origin of the custom. Perhaps drinking tea, with its caffeine, gives you heart.