Question

The use of “Delian healer” to refer to Apollo is called a(n) _____.

hyperbole
patronymic
epithet
motif

Answer 1

The correct option is:

Epithet.

An epithet is a byname, or a descriptive word or phrase, that accompanies or replaces a name after having entered common usage. It usually has different meanings when used to seemingly real or fictitious people, divinities, objects, and binomial nomenclature. At the same time, it´s possible that an epithet is used as a descriptive title, such as Pallas Athena and Alfred the Great.

The “Delian healer” refers to the role of Apollo as a healer in connection with Paean.

Answer 2

I am pretty sure that the use of “Delian healer” to refer to Apollo is called an epithet. I choose this figure because Delian was named so as the location of Apollo's birth is called Delos. Epithet definition says that it is known as descriptive title.
Hope you will agree with me.
Regards.