Closest cyclotomic integer to a cyclotomic number?

Short answer: Yes, there is such a function. Just map every element of your number field to $0$. Since you do not specify what properties your rounding function should have, you can't even complain that this answer is trivial.

As suggested by the remarks, you can choose an integral basis and round each element of the number field to the integer that is coordinatewise closest to this number. This also gives you many rounding functions, and you if you choose $1$ as an element of your integral basis then you may even pick a rounding function that restricts to the usual rounding function in the rationals.

The Euclidean minimum has little to do with the question; its image is a real number, most often rational, but rarely integral. You can of course map $\eta$ to some $z$ that minimzes the absolute value of the norm of $\eta - z$, but this is not well defined if your number field has nontrivial units.