It is already appropriate enough to describe this phrase as an epithet, of the sort with which the Homeric verses are rich. However, we might also call it a descriptive phrase. As it applies human attributes to aspects of nature, we could also say that it is an example of pathetic fallacy.
Pathetic fallacy is a conceit in which natural elements are afforded human motivations and attributes and are described as if they are human. In this instance, dawn is described as being human in that it has rosy fingers, whereas in actuality, this is only a figure of speech. We could also interpret this as a metaphor, where the clouds spreading across the early morning sky, limned in pink and gold, are represented as the fingers of dawn.
Consider what this expression does to dawn. What is the expression "rosy fingered dawn" giving to dawn? Hopefully you should be...
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thinking that it is giving it human characteristics, by referring to the "fingers" of dawn. Thus we can say that this figure of speech is an example ofpersonification. Of course, in the Greek world vision, Dawn in a sense was personfied anyway as a Goddess who brought morning into being, so perhaps this helps us to understand the way that dawn is personfied throughout this great epic classic.
However, just to remind you, personification is a figure of speech where an inanimate object is given human characteristics, in either actions or appearance. Note how the "rosy fingers" perhaps refer to the effect on the sky as the sun rises and how beautiful patterns appear.
In Homer's Odyssey, what figure of speech is "rosy-fingered dawn"?
In general, the line "When rosy fingered dawn appeared..." is first of all imagery. Imagery is an description that paints a mental image in the reader's mind.
Specifically, this is also an example of a specific kind of imagery known as personification. Personification is giving human characteristics to non-human things. The dawn, actually, is not made of fingers: fingers belong to people.
This could also be considered a metaphor. A metaphor is a literary device. Defined, a metaphor is the comparison of two dissimilar things with similar characteristics, as if they were the same.
In this example, the dawn is being compared to fingers.
All of these literary elements named above are known as figurative language. With figurative language, the reader does not take the description literally, but figuratively.
So the statement is considered figurative language. Imagery is an example of figurative language, and personification and the metaphors are examples of imagery (which is a kind of figurative language).