Why is it Euler's 'Totient' Function?

The Latin tot is correct as an origin for the root, but the suffix '-iens' doesn't originate with Sylvester either who was undoubtedly thinking of the already fully-formed word totiens when he coined 'totient.' Compare this to how quotiens enters into English as 'quotient.'

Sylvester knew Latin well enough that he would have been aware of the parallel between totiens and quotiens, which is actually a very manifest parallel since they function together as correlative conjunctions. A clause will introduce quotiens - how often; the next clause will answer totiens - this often.

ex: quotiens doces, totiens disce. 'Learn as often as you teach.' (literally, 'as often as you teach, learn this often.')

Correlative conjunctions like this are common in Latin. Here's another you'll recognize:

quantum - how much, tantum - this much.

Anyway, it seems to me that the word totient is meant to refer to the abstract notion of saying 'here is how many there are.' It doesn't seem to reference the quality of being relatively prime or any other quality.

(But speaking of 'qualities,' there's also qualis - what kind, talis - this kind, which hopefully goes to show how common these q-t correlatives are.)

It comes from the Latin tot--"that many, so many" (as in "total").

Going with a similar word: quotient

quotiens (how many times) = quot (how many) + tiens (times)

If totient has a similar origin, than it would mean "that many times" or "all the times". It probably refers to "all the numbers" coprime with $n$.

In latin "totus" means "all" or "whole" - see under the IE root teuta-