I'm just interested in knowing if there is any non-arbitrary basis for using prefixes "mono" or "uni" when words are initially being coined. As far as I can tell, they mean the same thing as a prefix. For example, why use "monochrome" instead of "unichrome"?
In a similar vein, what about other words, with prefixes like "bi", "di" or "duo"?
To be more specific, I suppose I wondered if there was any inherit connection between the prefix and its root word; like maybe if the root word is from Latin, the Latin prefix should be used, or something.
Best Answer
I’m gonna have to disagree with @prash, and so I’ll post this as a competing answer, offering an alternative take on The Rule:
The prefix mono- comes from Greek monos, itself rooted in the Proto-Indo European *men- (small). uni- comes from the Latin unus, itself from PIE oinos (and thus related to the Greek oinos, “ace on a dice”; Ancient Greek otherwise used alpha for the number one).
Forming words from mixed Latin and Greek roots used to be frowned upon. In the classical tradition, you would form things either from Greek or Latin, not mix them together.
As time evolved, a number of Greek and Latin roots entered the English lexicon, mostly as suffixes and prefixes. Because they became part of the language in that way, they could be combined with other words, being somewhat free of the older rules: you wouldn’t go forging a brand new words directly from Latin and Greek roots, but you would add a prefix (say, auto) that came from Greek but was now assimilated into the English language, and put it in front of an English adjective (say, mobile, which came from Latin movere through Middle French). And kaboom, you have: automobile. And few frown upon it.
So, what's the current status? It's now mostly a matter of taste. There is no strict rule, and you should form the word that sounds better to your ears.
Wikipedia calls words formed with roots of different languages hybrid words, though I've never really heard the term used. I’d prefer we call them heteroradicals (tip of the hat to JSBᾶngs).
Moreover, the Wikipedia article linked above has a long list of such English words, many of which are very common: homosexual, television, hyperactive, electrocution, bigamy, Minneapolis, etc. It even includes the following two words which have the mono- prefix:
TLDR — I usually don't do tldr’s, but this has turned into a general post on the topic while you had a very specific question. So, here's the answer to your very specific point: it's a matter of taste, choose the one that sounds best. If the roots are from the same language, as an added bonus it will please some more people.