My guess is that such alternate pairs arose either from words imported from different parts of China, or from different time periods, that gave the same character two different pronunciations.
Your observation is correct, and this statement is pretty much the answer.
What is the "official"/technical name for this phenomenon?
I don't know if there is a word for the differences in 音読み, but there is a name for each 音読み, and respectively they are 漢音、唐音、吳音, and for some Kanji there is even a 音読み called 慣用音. These represent the sounds of a Kanji imported in different time periods and from different parts of China. Henceforth the different 音読み for a single Kanji. Let's explore the 音読み's of the Kanji 明. (We'll only discuss 音読み and not 訓読み here)
漢音
The 漢音 for 明 is めい, and generally, 漢音 is the most common 音読み for most Kanji. 漢音 is imported during the 7-9 century, and it was during the 唐 dynasty, one of the most prosperous dynasties of China, and it was during this time the Japanese brought a LOT of cultural stuff into Japan. Even so, this reading is called 漢音 and not 唐音. 唐音 is something else.
唐音
The 唐音 for 明 is みん. It came during the 宋・元 dynasties, and generally anything after the 10th century is considered 唐音. 唐音 is the rarest reading among the three, and not every Kanji has it. Even for 明 which has a 唐音 of みん, I cannot think of any common word in which 明 is read as みん. For the 明 dynasty, it's read as みん though. 明朝{みんちょう}
吳音
The 吳音 of 明 is, as you have noticed, みょう. 吳音 is actually the oldest of the all, coming to Japan during the 5-6 century, and is mainly from the southern part of China. Even today, the Shanghai area is sometimes referred to as 吳 in Chinese, and the Shanghainese dialect is called 吳語 in Chinese. A lot of Buddhist words uses 吳音. For example, normally 利益 is read as りえき in daily life, but in a Buddhist context, you'd read the same word with 吳音, which is りやく.
慣用音
This refers the the 音読み that, after gotten into Japan, got corrupted and no longer resembled any original Chinese readings. We could say these developed naturally among the Japanese, but since it's a sound corruption of the original Chinese reading, it's still considered 音読み and not 訓読み. I could not find a 慣用音 for 明, but if you look up online you'll find 慣用音 for other Kanji's.
How should I know which one to use?
Sorry, but you just have to know them by rote. Each word has its own history, got imported from its time and therefore there is no consistent rule to govern them all. Even for the same word like 利益 there could be more than one readings, but there is always a reading that's the most prevalent, and most of the times you'll get it right by using the most common reading you know for that Kanji.