Are "money" and "mind" cognates?

Solution 1:

Yes, they're all related.

The Latin name Monēta comes from the verb moneō, monēre, monuī, monitus, 'to warn'. "Juno the Warner" was a cognomen of the goddess Juno; there's a story about geese in the link. The fact that the Roman mint was located (and named) at the temple of Juno Moneta is a coincidence. That's where a lot of names come from.

But, to return to the question, the etymology of moneō is very interesting. It turns out to be an O-grade ablaut variant of the PIE *men- 'mind' root, same as the other ones. But it's a Causative. Warn is a causative in PIE because it meant 'cause to think (of)'. And they had a way to indicate this.

PIE was highly inflected, like Sanskrit, and it had a causative aspect, which was a suffix -i- or -iy- before the person/tense inflection. The PIE would have been something like mon-iy-, and by Latin that had become mon-ē-.

This "yodated causative" -iy- suffix came down into English too, surviving as a fossil palatalization in words like drench, which comes from older drink-y-an and meant 'cause to drink'. Thus, at some point in English history, you could lead a horse to water, but you couldn't drench him.

Many English verbs that end in [tʃ] or [dʒ] can be paired off with an non-causative verb like drink/drench. For instance, milk/milch, dike/ditch, make/match, blank/blanch, wring/wrench, and the like (vowel changes happen often, but consonants tend to linger).