In both contexts, "lie in" somewhat describes location. 1. If I "lie in" bed, I am located in/on the bed. This does not have to mean that you are staying in bed late; it simply means that is where you are at the moment. Whoever told you that "lie in bed" must only mean "stay in bed late" was not correct. 2. In the context "It's origins lie in....," the origins are being described as being found in in "Five Billion Day." (I am unsure of what "Five Billion Day" means..... was this an event/holiday?) To sum it up, if I lie in the bed, I am found/located in the bed. If the origins lie in ______, then that is where the origins are found/located. Hope this helps! :) Source: myself, American English

It lies in =What you are talking about (some fact or idea) began in, comes from, started from, or originated (kind of like was born from) that time/event. The truth lies in what he says. (The truth comes from / is in - what he says). What lies ahead? (What will come from / start in - the future?)