If you are being asked to describe or analyze what could be meant by the figurative phrase "peppermint wind," you might consider that it likely references coolness. We know, from the line before, that a bird will stop here to rest from his flight so that he can "cool" himself in the wind that blows.
We often think of peppermint as producing a cooling sensation—if you eat a peppermint, you might be able to sort of feel a tingling on your tongue and even up into your nose. Peppermint contains menthol molecules that trick our brain into thinking that something is cold even when it is not. This is why peppermint is so often considered to be refreshing, perhaps even after a meal or snack.
So, when the speaker describes the wind as "peppermint" using a metaphor (a comparison of two unalike things where one is said to be the other), he references all of the things associated with peppermint: things like being refreshing and cooling. We can understand, I think, how these words can be descriptive of wind as well as of peppermint, and so it seems a particularly apt comparison. The wind cools and refreshes the bird like a peppermint would cool and refresh our mouths or breath.