There is no mini in minuscule.
If you spell one word meaning “very small” as miniscule, hold on for a minute.
And by minute, I don't mean the noun designating a bit of time and pronounced minnit. I mean the adjective meaning “very small” and pronounced my-noot. (Actually, in a sense, I mean them both, because ultimately they’re the same word.)
The first word in question is spelled minuscule, and the key syllable is minus- and not mini-, even though the latter seems to make sense by analogy with miniature, mini-van, mini-mouse, etc. Minus, of course, is “minor, lesser,” and -cule is a diminutive of that, making the lesser even less. (A “mini-scule” would be a tiny campus in Scotland, as scule is one recorded Scot spelling of school.) The ultimate source of minuscule is, of course, Latin, though we borrowed it from French. The earliest meaning of minuscule in English was its French meaning, the example of which has already appeared in this entry dozens of times. A minuscule is a lower-case letter, and the word eventually was extended to other things small.
Ultimately, though, you are safe to remember this spelling-bee guideline: to spell minuscule correctly, remember to “Hold on for a minute.”