The best way would be "a band of a friend of mine", formally. Informally, I'd use "a friend of mine's band" because you're possessivizing (it is now) "friend of mine". It also sounds "better".

It also seems to more closely align with the accepted rule of making possessive (happier?) other phrases where it makes more sense, like "Lewis and Clark's expedition" instead of "Lewis's and Clark's expedition".

Oh, here we go:

group possessive. You form the possessive for noun phrases by adding an ’s or an apostrophe at the end of the phrase: Jim and Nancy’s house, the Department of Chemistry’s new requirements, a three months’ journey. This construction gets cumbersome when the noun phrase is long, in which case you should probably use a prepositional phrase instead. Thus instead of saying the house that overlooks the bay’s property line, you should say the property line of the house that overlooks the bay.

(The American Heritage® Book of English Usage.
A Practical and Authoritative Guide to Contemporary English. 1996.)